Song is "Low" by Cracker, easy 4 chord song, just D C E G over and over.

I'm still learning how to do this stuff, but the key of G seems to fit best according to my understanding. The E is the "odd" chord, in that in the key of G, starting on E should give an Em triad (I think I'm saying that right?). Does that just mean that if one was going to play a (basic) solo over these chords, play in the key of G, but during the E chord stay away from C?

I'd be interested to hear what you guys have to say about this... obviously I've never done this before and I wanted to know if I'm even on the right track in my thought process. Thanks for any help / advice, I really appreciate it.
What you're doing is right, in terms of E being the odd chord. But playing those 4 chords right now I think that D is the tonic center. I would probably go with D mixolydian for improvising. E would still be the odd chord.
Yeah, you're doing it right.

The E is the odd chord, but the VI is very common in songs.

This makes the progression, in G Major, V - IV - VI - I, which, while it's not a super strong cadence, will work.

If I were to solo over this I would use G Major all the way through, but I would accent the G# during the E, and then resolve it to a chord tone (specifically G if I could) to end that phrase.
Quote by KoenDercksen
Tonal center is D in my ears...

So, D major with a G# on the E chord :

And C natural on the C?
D major being the key makes a lot of sense. you have Dmaj, obviously diatonic. Gmaj is the IV of the key. Cmaj is the bVII, borrowed from the parallel minor, and Emaj is V/V.
Anfangen ist leicht, Beharren eine Kunst.
Honestly, just mess around with the D pentatonic minor and you will be fine.
Thanks guys... the solo question is really more hypothetical than anything else... I'm just trying to put together some of what I've learned (or rather, what I think I've learned). I appreciate the input!