Happy Birthday is an easy tune to play because it uses just a few chords.
Here's a chord/lyric chart of Happy Birthday in the key of G
Happy Birthday is in 3/4 time, and written as a number chart it looks like this. We can see it uses the 1st, 5th, and 4th chords in the key. In the key of G, that's G, D, and C.
It should be easy to play Happy Birthday in any key! This will be good transposing practice. Here are some examples of the 1st, 5th, and 4th chords in some different common guitar keys:
G: G, D, C
E: E, B, A
A: A, E, D
D: D, A, G
C: C, G, F