The old town of Alcudia is surrounded by centuries old walls and is close to a wide choice of beaches. The town centre offers a great choices of restaurants and some nice boutique hotels. There is a twice weekly market.
A number of excellent villas in the upmarket areas of Barcares, Bonaire and Manresa are either on the coast or very near to it and are worth a look.
The Tramuntana Mountains are 20km away through the town of Pollenca via the Coll de Femenia or 10km away through Port Pollenca if you are aiming to go cycling on the newly resurfaced Cap Formentor route out to the lighthouse.
The Bay of Pollenca and the Bay of Alcudia are minutes away and Alcudia is a great choice to stay in when cycling in Majorca, especially if you are with others who will not be cycling as there is a lot for them to do as well.
The airport is about a far away as you can get from it on the island but it still is only around 50km away. Bike hire is readily available in the area.