One of my friends said that someone who denies or doesn’t believe and follow the teachings of the Church is still a Catholic. They basically said that you don’t have to believe everything that the Church teaches and has ever taught to be Catholic, but that you’re just a bad Catholic.

Obviously this is wrong, but I’m not sure how to respond to it. Where has any Pope said that Catholics must believe all that the Church teaches?

Furthermore, they said that one could not be a heretic until they were declared a heretic. I need Papal declarations that refute this claim as well. I have a hard time debating people because my words fall on deaf ears unless I present sources.

Quotes for you

2Corinthians 4:3-4

And if our gospel be also hid, it is hid to them that are lost, In whom the god of this world hath blinded the minds of unbelievers, that the light of the gospel of the glory of Christ, who is the image of God, should not shine unto them.

There is only one universal Church of the faithful, outside of which no one at all can be saved (Pope Innocent III, 1215).

We declare, say, define and pronounce that it is absolutely necessary for the salvation of every human creature to be subject to the Roman pontiff (Pope Boniface VIII, 1302).

Pope Eugene IV, Council of Florence, Sess. 8, Nov. 22, 1439, ex cathedra: “Whoever wishes to be saved, needs above all to hold the Catholic faith; unless each one preserves this whole and inviolate, he will without a doubt perish in eternity.– But the Catholic faith is this, that we worship one God in the Trinity, and the Trinity in unity... Therefore let him who wishes to be saved, think thus concerning the Trinity. “But it is necessary for eternal salvation that he faithfully believe also in the incarnation of our Lord Jesus Christ...the Son of God is God and man...– This is the Catholic faith; unless each one believes this faithfully and firmly, he cannot be saved.”

"If anyone in word and mind does not properly and truly confess, according to the Holy Fathers, all, even to the last portion, that which has been handed down and preached in the Holy, Catholic, and Apostolic Church of God . . . let him be anathema." - Pope St. Martin I

Pope Pius IX, Vatican I, Sess. 3, Chap. 3, 1870, on Faith:

Further, by divine and Catholic faith, all those things must be believed which are contained in the written word of God and in tradition, and those which are proposed by the Church, either in a solemn pronouncement or in her ordinary and universal teaching power, to be believed as divinely revealed.

But, since "without faith it is impossible to please God" [Heb 11:6] and to attain to the fellowship of His sons, hence, no one is justified without it; nor will anyone attain eternal life except "he shall persevere unto the end on it" [Mt 10:22; 24:13].

Pope Paul III, Council of Trent, Sess. V, 1546: “…our Catholic faith, without which it is impossible to please God [Heb. 11:6]…”

Sess. VI, Canon II: “If anyone shall say that divine grace through Christ Jesus is given for this only, that man may more easily be able to live justly and merit eternal life, as if by free will without grace he were able to do both, though with difficulty and hardship: let him be anathema” (Denz. 812).

Syllabus of Errors:

17. Good hope at least is to be entertained of the eternal salvation of all those who are not at all in the true Church of Christ. —Encyclical “Quanto conficiamur,” Aug. 10, 1863, etc.

Pope Benedict XIV, cuм Religiosi (# 4), June 26, 1754: “See to it that every minister performs carefully the measures laid down by the holy Council of Trent… that confessors should perform this part of their duty whenever anyone stands at their tribunal who does not know what he must by necessity of means know to be saved…”

Pope St. Pius X, Acerbo Nimis (# 2), April 15, 1905: “And so Our Predecessor, Benedict XIV, had just cause to write: ‘We declare that a great number of those who are condemned to eternal punishment suffer that everlasting calamity because of ignorance of those mysteries of faith which must be known and believed in order to be numbered among the elect.’”

Pope Gregory XVI, Mirari Vos (# 13), Aug. 15, 1832: “Now we consider another abundant source of the evils with which the Church is afflicted at present: indifferentism. This perverse opinion is spread on all sides by the fraud of the wicked who claim that it is possible to obtain the eternal salvation of the soul by the profession of any kind of religion, as long as morality is maintained. Surely, in so clear a matter, you will drive this deadly error far from the people committed to your care. With the admonition of the apostle, that ‘there is one God, one faith, one baptism’ (Eph. 4:5), may those fear who contrive the notion that the safe harbor of salvation is open to persons of any religion whatever. They should consider the testimony of Christ Himself that ‘those who are not with Christ are against Him,’ (Lk. 11:23) and that they disperse unhappily who do not gather with Him. Therefore, ‘without a doubt, they will perish forever, unless they hold the Catholic faith whole and inviolate (Athanasian Creed).

Pope Gregory XVI: “Therefore, they must instruct them in the true worship of God, which is unique to the Catholic religion.” (Summo Iugiter Studio # 6, May 27, 1832)

Pope Leo XIII (1902): “By his (Christopher Columbus’) toil another world emerged from the unsearched bosom of the ocean: hundreds of thousands of mortals have, from a state of blindness been raised to the common level of the human race, reclaimed from savagery to gentleness and humanity; and, greatest of all, by the acquisition of those blessings of which Jesus Christ is the author, they have been recalled from destruction to eternal life.” (Encyclical, Quarto Abrupto)

Pope Leo XII, Ubi Primum (# 14):

“It is impossible for the most true God, who is Truth itself, the best, the wisest Provider, and the Rewarder of good men, to approve all sects who profess false teachings which are often inconsistent with one another and contradictory, and to confer eternal rewards on their members… by divine faith we hold one Lord, one faith, one baptism… This is why we profess that there is no salvation outside the Church.”

St. Alphonsus Liguori, Sermons (c. +1760): “How many are born among the pagans, among the Jєωs, among the Mohammedans and heretics, and all are lost.”

"... these modern incredulous: and if ever someone had remained blinded by their sophisms, that he open his eyes to recognise the truth of our Holy Faith, absent which there isn't hope of salvation."

Preparation For Death: “How thankful we ought to be to Jesus Christ for the gift of faith! What would have become of us if we had been born in Asia, Africa, America, or in the midst of heretics and schismatics? He who does not believe is lost. This, then, was the first and greatest grace bestowed on us: our calling to the true faith. O Savior of the world, what would become of us if Thou hadst not enlightened us? We would have been like our fathers of old, who adored animals and blocks of stone and wood: and thus we would have all perished.”

St. Francis Xavier, Dec. 31, 1543: “There is now in these parts [of India] a very large number of persons who have only one reason for not becoming Christians, and that is that there is no one to make them Christians. It often comes into my mind to go round all the Universities of Europe, and especially that of Paris, crying out everywhere like a madman, and saying to all the learned men there whose learning is so much greater than their charity, ‘Ah! What a multitude of souls is through your fault shut out of heaven and falling into hell!’…

St. John Vianney: "Christians who are damned will suffer torments infinitely more rigorous than the infidels. The reason is that these strangers will be damned because they never heard talking about Jesus Christ and his religion; that they lived and died in ignorance."

St. Robert Bellarmine: "...no one is justified without faith in Christ."

St. Augustine (+428): “… God foreknew that if they had lived and the gospel had been preached to them, they would have heard it without belief.”

St. Thomas Aquinas, Sent. III, 25, Q. 2, A. 2, solute. 2: “If a man should have no one to instruct him, God will show him, unless he culpably wishes to remain where he is.”

Pope Pelagius I, Vas electionis: "That I may define by profession my faith, attached below, in which, by God's grace, it may be manifestly clear that I follow in the footsteps of the correct doctrine of the Apostles and Fathers[..]I confess that all men from Adam, even to the consummation of the world, having been born and having died with Adam himself and his wife, who were not born of other parents, but were created, the one from the earth, the other, however, from the rib of the man will then rise again and stand before the Judgment seat of Christ, that every one may receive the proper things of the body, according as he has done, whether it be good or bad; and indeed by the very bountiful grace of God he will present the just, as vessels of mercy prepared beforehand for glory, with the rewards of eternal life; namely, they will live without end in the society of the angels without any fear now of their own fall; the wicked, however, remaining by choice of their own with vessels of wrath fit for destruction, who either did not know the way of the Lord, or knowing it left it when seized by various transgressions, He will give over by a very just judgment to the punishment of eternal and inextinguishable fire, that they may burn without end. This, then, is my faith and hope, which is in me by the gift of the mercy of God, in defence of which blessed Peter taught that we ought to be especially ready to answer everyone who asks us for an accounting[..]But whosoever will hold, believe, or preach otherwise, him the holy and universal Church of God anathematizes."

Pope Benedict XIV, Apostolica (# 6), June 26, 1749: “The Church’s judgment is preferable to that of a Doctor renowned for his holiness and teaching.”

Errors of the Jansenists, #30: “When anyone finds a doctrine clearly established in Augustine, he can absolutely hold it and teach it, disregarding any bull of the pope.”- Condemned by Pope Alexander VIII

Pope Pius XII, Humani generis (# 21), Aug. 12, 1950: “This deposit of faith our Divine Redeemer has given for authentic interpretation not to each of the faithful, not even to theologians, but only to the Teaching Authority of the Church.’”

St. John Chrysostom:

“Lest anyone say ‘What about those who do not believe in Jesus Christ’ listen to what Jesus says:

‘I know mine and mine know me’.”

-John 10:14

What denying this doctrine leads to:

Abp. Lefebvre, Sermon at first Mass of a newly ordained priest (Geneva: 1976): “We are Catholics; we affirm our faith in the divinity of Our Lord Jesus Christ; we affirm our faith in the divinity of the Holy Catholic Church; we think that Jesus Christ is the sole way, the sole truth, the sole life, and that one cannot be saved outside Our Lord Jesus Christ and consequently outside His Mystical Spouse, the Holy Catholic Church. No doubt, the graces of God are distributed outside the Catholic Church, but those who are saved, even outside the Catholic Church, are saved by the Catholic Church, by Our Lord Jesus Christ, even if they do not know it, even if they are unaware of it...”

Pope Clement V, The Council of Vienne, 1311-1312: “Besides, only one baptism regenerating all who are baptized in Christ must be faithfully confessed by all just as ‘one God and one faith’ [Eph. 4:5], which celebrated in water in the name of the Father and of the Son and of the Holy Spirit we believe to be the perfect remedy for salvation for both adults and children.”

Pope Clement V, The Council of Vienne, 1311-1312: “But since one is the universal Church, of regulars and seculars, of prelates and subjects, of exempt and non-exempt, outside of which absolutely (omnino) no one (nullus) is saved, one is the Lord, one is the Faith and one is the baptism of all.”

Pope St. Leo the Great, Letter 105, May 22, 452: “… giving thanks to the Merciful and Almighty God that He has suffered none save those who loved darkness rather than light to be defrauded of the gospel-truth.”

Fr. Francisco de Vitoria, O.P., a famous 16th century Dominican theologian,

“When we postulate invincible ignorance on the subject of baptism or of the Christian faith, it does not follow that a person can be saved without baptism or the Christian faith. For the aborigines to whom no preaching of the faith or Christian religion has come will be damned for mortal sins or for idolatry, but not for the sin of unbelief. As St. Thomas says, however, if they do what in them lies [in their power], accompanied by a good life according to the law of nature, it is consistent with God’s providence that he will illuminate them regarding the name of Christ.

Pope St. Siricius (A.D. 385): “… we also say that to infants who will not yet be able to speak on account of their age or to those who in any necessity will need the holy stream of baptism, we wish succor to be brought with all celerity, lest it should tend to the perdition of our souls if the saving font be denied to those desiring it and every single one of them exiting this world lose both the Kingdom and life. Whoever should fall into the peril of shipwreck, the incursion of an enemy, the uncertainty of a siege or the desperation of any bodily sickness, and should beg to be relieved by the unique help of faith, let them obtain the rewards of the much sought-after regeneration in the same moment of time in which they beg for it. Let the previous error in this matter be enough; [but] now let all priests maintain the aforesaid rule, who do not want to be torn from the solidity of the apostolic rock upon which Christ constructed His universal Church.” (Decree to Himerius on the Necessity of Baptism)

Second Council Of Constantinople, Sess. 3, AD 553: “… the holy, catholic and apostolic church of God, if anyone separates himself from its communion by holding contrary opinions, such a person, since he alienates himself from the orthodox faith and numbers himself with the heretics, is justly condemned and anathematized by the holy Church of God.”

Here is Canon 1325 of the 1917 Code of Canon Law: (emphasis is mine)

Canon 1325 (1983 CIC 209, 751, 755)
§ 1. The faithful of Christ are bound to profess their faith whenever their silence, evasiveness,
or manner of acting encompasses an implied denial of the faith, contempt for religion, injury to
God, or scandal for a neighbor.
§ 2. After the reception of baptism, if anyone, retaining the name Christian, pertinaciously
denies or doubts something to be believed from the truth of divine and Catholic faith, [such a one
is] a heretic; if he completely turns away from the Christian faith, [such a one is] an apostate; if
finally he refuses to be under the Supreme Pontiff or refuses communion with the members of the
Church subject to him, he is a schismatic.

And the previous canon explains what constitutes something to be believed as divine and Catholic faith:

Canon 1323 (1983 CIC 749–50)
§ 1. All of those things are to be believed with a divine and Catholic faith that are contained in
the written word of God or in tradition and that the Church proposes as worthy of belief, as divinely
revealed, whether by solemn judgment or by her ordinary and universal magisterium.

Many "Catholics" don't believe in contraception being a sin for example, so I guess they're not Catholic? If I'm wrong, please correct me.

In one sense, if baptized Catholic, and follow some or most, but not all the Church teaches, then they’ll be judged as Catholics, but unless they’ve sincerely Confessed, they can be said to be fallen Catholics, schismatic, heretic, or apostate if they’ve repudiated the Faith entirely.
It is better to have never been Catholic at all, never Baptized Catholic than knowingly turned away from the teachings of the Faith.
As for those Baptized, but ignorant, not through their any fault of their own, that’s up to God. Pray for such persons to be enlightened and converted. Even if they’ve died, we don’t know the heart, whether in Purgatory or lost. Our prayers aren’t wasted because Our Lady directs them to those most in need.