All of the previous answers here are really good, and alongside G, D, C, A, and Em, I feel you should also learn Am, as well as a few easy jazz chords to spice up your sound. Because I can't post images of the chord charts, try searching each of the chord names on www.jguitar.com to see charts of how to finger each chord.
Playing in the key of G will make G, Em, C, D, and Am useful to you. Playing in C will make C, G, Em, and Am useful (and is a good opportunity to learn Dm). Playing in D will make D, G, A, and Em useful. Additionally, learning E major is good because it's the lowest open chord on standard tuning; a lot of rock music is played in the key of E because all of the open strings appear in this key.
Now, if you know where to use them, there are some extremely easy jazz chords you can learn, such as A major 7th, E minor 7th, G 6th, and C major 7th. You should start by learning songs that use these, and then see how they sound in relation to other chords. Try playing Em, then Em7, and listen to the difference. Try the same with G and G6, A and Amaj7, C and Cmaj7, etc. Their dissonance makes them unsuitable for use on their own but they can sound absolutely beautiful in between more "normal" chords. The G6 chord appears in the key of C, the Amaj7 appears in the key of A, the Cmaj7 appears in the keys of C and G, and the Em7 appears in the keys of C, D, and G.
Another chord that will be infinitely useful when practiced is F, but this one is harder than the others I've mentioned, as it's a bar chord. You can practice this further down the neck, closer to the body of the guitar, where less force will be required to fret the strings with your index finger (this is a hint as to why F, and later, the much harder B, is so important). Learning F and being able to play it reliably is the foundation to playing just about any chord and any song you want.