What are the best things to see and do?
Been there a couple of years ago, and from what I recall:
1. The market place and old town around it
2. The city walls
3. The Gotland museum
4. The seaside walk
5. All the church ruins
Visby is a small town, you can do most things in a day. The ice cream shop „glassmagasinet“ near the harbor with their about 100 flavors is another fun thing, and there was a medieval themed restaurant, plus some small craft run restaurants, cafes and ice cream shop around the market.
If you are there for more than a day, consider renting a bike for a trip along the coast, or a bit longer by bus or boat.