The old town is small and steep — you'll walk everywhere within it. Wear shoes with grip; the marble of the staircases is slick when wet.
For trips along the coast — Igalo, Meljine, Savina monastery — there is a frequent local bus that runs the length of the seafront, and the Pet Danica promenade is usually pleasanter on foot anyway. Taxis are cheap and metered.
For day trips across the bay to Žanjic or the Blue Cave, the small boats from the promenade are the obvious choice and run on demand through the summer. A rental car opens up the wider region — Mount Orjen, Lovćen, Croatia — but parking in the old town is limited and on a steep slope.