One thing I'd recommend is simply to get lost.

Move away from the 3-4 places the tourists cram into, and simply walk around. The Ghetto - the "original" Jewish ghetto - is particularly interesting. Get over to the Giudecca island. Maybe even go see a football match if the local team is playing, it's quite an interesting atmosphere. If you're an early riser, a ride on the No. 1 Vaporetto from the station to Giardini along the Canal Grande is amazing.