I am attempting to put in a new bathtub after gutting our old bathroom. I am not sure if the drywall surrounding the tub should go down to the floor or stop at the tub?

Should stop at the tub. Most tubs have a flange that goes against the studs. Might want to use cement board not drywall. Check out this link.

Tile Showers and Tubs (http://www.thetiledoctor.com/installations/st_tubs.cfm)