The intercity buses are run by the Transport de les Illes Balear; these are regular, comfortable and cheap.
Search for TIB Mallorca on the web to find the current timetables (the bus service is usually punctual). Note that bus frequency drops to about 50% during weekends and holidays, so plan ahead. Most buses take at least two bicycles in their luggage compartment and start off from the main terminal under Plaza d'Espanya
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.