Public transport in Mallorca basically consists in buses. The city is well communicated by buses, but unfortunately rather expensive and bad communicated with the rest of the island.
Palma’s buses work enough well and nearly every place can be easily reached by taking one or two buses. The fare for each bus you take inside the city is about 1.10 Euro and by now there is no bonus for buying a set of tickets.
The main lines are number 15, which goes from the centre to El Arenal and number 3, which goes from El Pont d’Inca to Illetas. There is also a line that goes from the Airport to the city centre, but it is not recommend as it is slow and not reliable.
Depending on which number you take, it will come from every 7 minutes to every 30 minutes. This can be easily found at any bus stop.
Also you should keep in mind that during the night there is no bus service. Some of the routes stop at 23:00 and the latest such as the Arenal one (number 15) do the last at 01:00, so keep it in mind if you are going on party at night.
Some rules you may follow, despite of what you think, are: you are not allowed to go in to any public transport wearing bath clothes, without any shirt or t-shirt, eating or smoking, and you may give the driver small coins or notes (up to five euro is fine).
Transport to other parts of the island is another history. The services start from the streets around the Plaza de España, in the centre. There are only a few buses every day to the main areas of the island: you should check the timetable and be on time if you do not want to be left and have to wait until the next day to get yours.
To find the timetables you can visit this page where you will find a list containing all of the routes for inland transportation run by the TIB.