For moving around Palma i would recommend you to rent a car, its the easiest way to see the whole island. Taxi is quite expensive, its only worht to use it in the city. There is bus connections from the city to the nearby beaches like Palmanova, C´an Pastilla, S´Arenal. I would recommend you to take the train to a town called Soller. Its a slow train that goes through the mountains, and gives you beautiful views.
Apart from this, most beaches have scooter, Bicycle and Rollerblades rent, which is quite good to move around in the beach towns.

