Yes, you may hear "dueling bands" (classical music) in St. Mark's at night. It's really pretty there and all around Venice at night. I'd just walk around at night, take a vaporetto ride up the Grand Canal. I can't think of a particular spot - just find a place to eat or have a drink (maybe in St. Mark's square, maybe elsewhere).
The main "tourist trail" in Venice is the street(s) from the train station through to the Rialto Bridge (much of this is called "Strada Nuova") and from there to San Marco (St. Mark's). Much of that street can be mobbed with tourists all day. But if you wander up into the ghetto area (north of the train station) for example it should be less crowded.
On my last visit to Venice last May, I did a great morning walk from the train station to Rialto - but not along the touristy Strada Nuova. instead, I walked south of the Grand Canal through the San Polo neighborhood, through a maze of small squares and quiet streets. But you shouldn't get lost - just follow the signs "Per Rialto." In the morning - really a great time to explore Venice in my view - there were almost no tourists in this stretch, just locals taking their kids to school or walking to work, shopkeepers sweeping in front of their stores or chatting with each other, people carrying in supplies to the shops and restaurants from boats. It feels like a very different Venice from "tourist Venice."