Mental prayer is very easy. It is talking to God, Our Lady, a saint(s), etc. It is just talking and thinking as we talk. In this way it differs from formal, written " vocal" prayers. The subject matter about which we speak should be such that it elevates our minds to God and makes us want to grow in understanding of His teachings, grow in the practice of virtue, etc. The goal is to help us become holy by interacting with God on a personal level. Formal prayers can be said by a group. Mental prayer is made by individuals, even if they all make mental prayer at one time, as monks and nuns do. Each soul is individual; each one's prayer is particular to that individual.
God often offers us an invitation to mental prayer while we are praying vocally. Example: We are saying the Rosary and as we think about a mystery there is something we wish to understand better. We can stop and just talk to God or Our Lady about what we wish to understand. If we feel pity for Jesus in His Passion, tell Him. THAT is mental prayer. Tell Him all you feel. The same goes for any other mystery, or any truth of the faith such as His Presence in the Holy Eucharist, or any dogma like His Birth, etc. If you've done spiritual reading, you can talk to Him about what you read. Ask Him to guide you to find subjects which will hold your attention and you will enjoy thinking about and talking about with Him. The same goes for Our Lady.
You can imagine yourself present at any incident in the Scriptures. What would you do? If you were present when Jesus gave a parable, and you had the chance to ask Him a question, what would you say? This is mental prayer. Ask the question!
What if you were the Samaritan woman at the well. What if you found Jesus all by Himself and He asked you for a cup of water. What would you want to talk to Him about if you had time with Him all alone. What if you were Nicodemus coming to Him in the night as He prayed alone on the mountainside. What would you want to say to Him? Say it. This is mental prayer.
We say what we feel in our heart and we say it to whomever we are praying to. Then we spend time thinking about what we said. This time of silence gives the Holy Spirit an opportunity to inspire us, or enlighten us, or even just give our souls more peace than we had before we prayed.
Even if you have a problem to talk to Him about that isn't particularly religious in nature, make it so. Talk to Him about the problem and then think about what He taught in the Gospels--can you find any words of advice there that pertains to what is bothering you? Talk to Him about it. Elevate even the most mundane things to a spiritual level this way so that every moment of your life becomes spiritualized so to speak, and therefore more for the glory of God.
Don't worry now about contemplation if you are a beginner. That usually comes later. There are two kinds: acquired and infused, but unless you receive a special grace from God drawing you to that kind of prayer (it just happens; you can't refuse so don't start wondering if you have it or not), you begin with simple meditation and when your mind wants to stop and think about something, you do so and talk to Our Lord or Whoever is the subject of the meditation. It is a conversation with the One Whom we know loves us, as St. Teresa of Avila says. It's that simple. When St. Margaret Mary asked her Mistress of Novices how to make mental prayer, she was told "Go place yourself before God as a blank canvass before the painter." What was meant was, just go to Him and let HIM do with your soul what He wants to do. It may feel awkward at first but if you truly believe you ARE talking to God or Our Lady or a saint, you will become comfortable with it and find delight in it.
I recommend the method of meditation taught by St. Francis de Sales. St. Teresa of Avila used to like to think of Jesus in His Agony in the Garden. She wrote that it is good to use a good Life of Christ for reading and that this helped her with her prayer until God raised her to a higher state of prayer. Her works talk about how to make prayer and they are excellent.
The saints recommend meditating on the Passion all year long because no matter how deeply you go into thinking about it, there is still more to understand than we possibly can be given this side of Heaven and it consoles Jesus for us to think of what He suffered for our own sins and helps keep us from sinning more if we remember what He endured because of them.
What I've written is a very short compilation of years of study on the spiritual life from the writings of the greatest saints and theologians. It works.
And remember: Of ourselves we can do nothing but sin. If you feel drawn to mental prayer, it is God drawing you to it. Accept the invitation. He will help you do what He wants of you. Don't be scared. It would be impossible to find another person who understands you so well as your Lord and Savior. Mental prayer can be a sheer delight.
Read some books about the spiritual life to understand more about this if you wish. Many are on archive.org. for free.