The old town, cathedral, Alcazaba, and Cable Inglés are all walkable, though the climb up to the Alcazaba is exposed and hot in summer. Start early if you are visiting in July or August.
Local buses cover the beach districts and the airport route. Taxis are easy to find around the station, port, and central squares.
For Cabo de Gata and the province's inland desert scenery, hire a car. Public transport reaches some villages, but it is too thin for relaxed exploring.