I'm writing a song for a project and putting piano chords under the melody. I don't know if this makes sense but for example, if I'm in C major, and there is an "E" in the voice, should I pair it with a root position I chord (CEG) or should I have a I6 chord (EGC) because the voice is singing an E?
In popular music, the most common chord positions are the root and the second inversion. First-inversion chords tend to sound rhythmically "weak". In any case, doubling the third of a major chord between the bass and treble is probably not a good idea, unless you really want that sound for some reason. Try it, and some alternatives, and use your ears!
If your bass line follows the melody "exactly", it's not really functioning as the bass line - it's more like somebody trying to sing along with tune, but a few octaves too low. If your arrangement sounds "OK" with just the tune and the bass line and nothing else, that's usually a good starting point.
For chords at the end of the piece (or really important sectional endings) a root position chord is often best. Other places, the choice of root or first inversion should be (at least in my opinion) which one makes the bass line sound best (in the composer's opinion.) Usually (at least in chorale style with 3 or 4 voices) one prefers not to double thirds of major chords; doubling thirds of minor chords (ii,iii,vi) is fine. I'm not sure of the reason but it might be that a major chord with a doubled third may be mistakenly heard as a Neapolitan Sixth.
In more popular styles, less worry is needed about doublings or parallels. If it sounds good, it's fine. (That's why I like the idea of having a smooth bass line.