Excellent question! I try to pray a Litany related to the specific month, i.e. this month is the Litany to the Sacred Heart, next month is The Precious Blood. Get a good Traditional Catholic calendar and you will know which Litany you can pray as well as various novenas. Hope that helps!

Still quite new to the faith and am unfamiliar with many of the smaller things. For example monthly devotions I had no idea until May this year. What devotions does the Church have each month of the year?

January
February
March
April
May - Mary
June - Sacred Heart
July - Precious Blood
August
September
October
November
December

From my memory, in addition to what is post above, January was devoted to the Holy Family, though I notice some sources devote January to the Holy Name of Jesus and February to the Holy Family.  March was devoted to St. Joseph.  May, June, and July as above.  October is the month of the Rosary and November is devoted to the souls in Purgatory.

The Blessed Be God prayer book and missal (1925) has this list:
 January - The Holy Name of Jesus
February - The Holy Family
March - St. Joseph
April - The Holy Ghost
May - The Blessed Virgin Mary
June - The Sacred Heart
July - The Precious Blood
August - The Most Pure Heart of Mary
September - Our Lady of Sorrows
October - The Queen of the Holy Rosary
November - The Souls in Purgatory
December - The Immaculate Conception of the Blessed Virgin Mary

Thank you for posting Miseremini.

I had looked in my copy of My Prayer Book by Fr. Lasance, but I was looking in the index and didn’t quickly find the list there, Blessed Be God had an index listing for Devotions for Every Month.
 
The two lists are nearly identical, except for September when Fr. Lasance has the Holy Crosss, which is appropriate as September 14 is the Feast of the Exaltation of the Holy Cross.  But again, September 15 is the Feast of the Seven Sorrows, and the two are very connected.
 
Both Blessed Be God and My Prayer Book are available in reprint, but one might find a better priced used copy (original edition) on eBay (I was very fortunate to find both there).  Both have prayers, ejaculations, novenas, litanies for each month, both are excellent.  Either would be a good choice for the OP.