Agh! Where do I start?
The most typical pieces you’ll hear at Mass are the Kyrie, Gloria, Sanctus, Agnus Dei: these are used in many places in our own diocese.
The most important pieces to learn in terms of being able to sing in solidarity with other Catholics are the Pater Noster (of course), the Credo, and the four Marian hymns which are noted below (especially the Salve Regina). These are frequently sung in large gatherings of Catholics for Mass or other events.
Some of the hymns have repeating melodies, making them easier to learn. Two fairly simple ones, and good ones to know, are Adoro te Devote and Veni Creator Spiritus.
Chants of the Ordinary (of the Mass)
Kyrie XVI (Lord Have Mercy)
Gloria VIII (Glory to God in the Highest)
Credo III (Creed)
Sanctus XVIII (Holy Holy)
Mysterium Fidei (the Mystery of Faith)
Pater Noster (Our Father)
Agnus Dei XVIII (Lamb of God)
Verbum Domini (The Word of the Lord)
Mysterium Fidei (The Mystery of Faith)
Ite Missa est (Go forth, the Mass is)
Adoro te Devote
Alma Redemptoris Mater
Marian Hymn sung from the First Sunday of Advent to the Presentation
Ave Maris Stella
Ave Regina Caelorum
Marian Hymn sung from the Presentation to the Easter Vigil
O Salutaris Hostia
Notation available in missallette (should be familiar)
Pange Lingua / Tantum Ergo
Marian Hymn sung from the Easter Vigil to Pentecost
Marian Hymn sung from the day after Pentecost to the first Sunday of Advent
Veni Creator Spiritus