Palma has an extensive, cheap, clean (gaz powered) and efficient public transport network.
Their great website is available in English and will help you find your way around town remarkably easily.

The open top double decker tourist buses (No 50) are a quick and easy way to get a look around town (they also go up to Bellver Castle). The Bus de Nit (Night bus, No 41) links Plaza d'Espanya with the nightclubs on the Paseo Maritimo, from midnight until 7 am.
The Airport bus (No 1) costs 2€,{at the time of writing} and runs every 20 minutes at peak times. Note that the airport bus runs along the Paseo Maritimo but does not pass in front of the Cathedral; it goes around the old town instead along the Avenides.