If you want to stay in Alcudia, George's Casa is the place to be. Though not very spatious, the roof terrace and the complete house to your possession, it feels much bigger. Stepping out your frontdoor you immediately enter the mediaval world of the old town. In the morning, watching all the tourists enter town from your balcony, you will feel like a local. George and his wife have done a great job with the house and have a keen eye for nice details. If you come for beaches you can do a walk, but in summer it's hot and probably easier to use your car if you rented one (which I would advice). If you use the right parking lot, just outside the city walls, it costs you only 4 minutes to get it. Check the St Joan festivitys and you might be as lucky as we were to watch the fireworks from the roof top of the casa.
Definitely enter the Tramontana Mountains and turn left after Lluc's Monestery (not worth the wile) to descent the narrow road to the 2 small beaches.