Palma is the only centre with its own local transport system, EMT. Buses are the main way around, although a metro line (of more interest to commuters in the suburbs than to visitors) runs from the centre to the university.
It’s easy to get around Palma (especially the old centre) by bicycle, although cycling lanes are limited (the main one runs along the shoreline).
Palma is well supplied with taxis, with several stands around the city. Elsewhere on the island, you may not necessarily find them waiting when you need them, but generally they’re fairly easy to order by phone; ask at your hotel or local tourist office.