The old town inside the walls is closed to most car traffic and is small enough to cross on foot in under twenty minutes — walking is the only sensible way to see it. The cobblestones are punishing on suitcases and high heels; bring sensible shoes.
For the rest of Gotland, you really need either a car or a bicycle. The flat terrain and minor roads make cycling a genuine pleasure, and the rental shops near the ferry terminal and the city wall do brisk business. Local buses cover the main villages and the better beaches but run infrequently. Taxis are available but expensive — Gotland is a long, thin island and even a "short" trip can run to 30 km.