Palma is expensive. I have stayed there on a number of occasions. Maybe don't stay in Palma for the week.
Consider Port de Pollenca or Soller for maybe 5 nights and Palma for 2. Quite often flights arrive late in to Palma.
When you arrive at Palma airport you can get the bus outside arrivals, across the bus park to the stop beside the outside escalators and under the cement awning. Bus no 1.
This bus runs along the coast, then in to the town and then back down again to the coast. If you are travelling inland to other locations or other coastal locations you get the regional buses at Placa de Espana. The airport bus stops here. Both the train station and bus station are located under the park at Placa de Espana. Great bus station. Buses leave from their bays with good signage. You will pay max €8 per person to travel to the other side of the island.
Port de Pollenca is beautiful. Day trips from there to Pollenca old town or Alcudia old town or on to Alcudia beach. We stayed at Hoposa Apartments Daina.
Soller is smaller and also beautiful. You can get the bus here or the old cowboy style wooden train from the nearby station. This is a trip in itself. You can go to old town Soller or onwards on an even narrower wooden train to the Port of Soller.
http://www.mallorca-now.com/palma-soller-train.html