Corban W. answered 01/27/20
Music, Mandolin, Guitar, Songwriting, Music Theory, ESL
C, F, and G7 are the 1, 4, and 5 chords in the key of C. They are also the three chords in that key that contain major triads.
A triad is a three note chord built using intervals of thirds. G7 is a four note chord that is an extension of a basic G triad.
Every key has three diatonic (natural to the key) major triads, three diatonic minor triads, and one diminished triad.
So, in your case, you are already using the major triads. There are a lot of great songs that use just these. All the minor triads in C are still available and are good options. These are: Am, Dm, and Em.
Common non diatonic chords choices would be A, D, E, Bb, and Eb (the explanation for why these work well and how to use them is a bit longer, but would be easy to explain conversationally in a lesson).
Hope this helps,
Corban