I've finally gotten onstage and starting playing in a band. In the tablature that they give out, many of the chords are some note "over" a chord. Such as F#/D or F#/G for a song in D. What's going on...

You've picked a difficult one to try to learn with! 🙂 The verse is straightforward, it's in the key of D major. D, A and Bm all belong to the key of D. The chord sequence used here isn't a common one...