Port de Pollenca is a good recommendation, it is a small port town with a pleasant beach and waterfront. You don't state when you'll be going as the time of year is a big factor on the island.
With a car you'll be nowhere more than 1.5 hours drive from one point on the island to another. Close to Port de Pollenca is the old walled town of Alcudia (theres also Port de Alcudia but it's a bit touristy and full of resorts) which is interesting and fits the bill for a local feel. Further down is Arta which will give you a very good local experience and the nearby caves are stunning. North and west from Port de Pollenca are the charming towns of Soller, Valldemosssa and (quieter) Deia. These can be a bit busy during high season, get there early or later for less crowds.
Down in the south and southwest are the delightful beach villages and ports of Cala Pi, Colonia de Sant Jordi, Cala Santanyi and Portopetro. Close by are the towns of Santanyi, Campos and Llucmajor. Campos in particular has a very good market on Thursdays and Saturdays. It has a slow, rural atmosphere but it's very enchanting as is Santanyi. The popular and beautiful unspoilt beach of Es Trenc is nearby. You won't get the cruise ship tourist hordes bussed in to towns in the south unlike Palma, Valldemossa and Soller which are the perennial favourites.
https://www.seemallorca.com is a worthwhile and comprehensive website. There is lots to see and do, I've been visiting for years and I'm still discovering places and sights.