Relaxed is not a word that comes to mind when thinking of Palermo, so if you’d like a relaxed holiday, don’t base in Palermo.
That said however, Monreale is best reached by bus from Palermo. It’s best to avoid driving in Monreale, or even Palermo’s ring road to Monreale. And although AMAT has discontinued Palermo’s urbano bus service from Palermo to Monreale, AST runs bus service, and City Sightseeing Palermo runs service from May to October.
You had mentioned that you were arriving in Palermo, not interested in touristy destinations, and that you wanted to enjoy some hiking, beach, local cuisine, and ruins.
Cefalù is a resort town with a nice beach and good restaurants, and also hiking possibilities up La Rocca or in the nearby Madonie.
You could visit Monreale via Palermo on public transport (train or bus) from Cefalù.
You could also hire a car and take day trips to Piazza Armerina’s Villa Romana del Casale and Agrigento’s Valley of the Temples from Cefalù, but overall Cefalù isn’t the most convenient base for exploring ruins, nor is it convenient for trekking Lo Zingaro, which you would like to do.
Palermo’s airport is located west of the city, and with an arrival at Palermo’s airport, you may find Western Sicily more convenient for ruins (Segesta, Selinunte, Cave di Cusa, Eraclea Minoa, the Valley of the Temples....), Lo Zingaro, beaches....
You could hire a car at the airport on arrival and explore on your own. You could also take a tour, or prearrange a licenced guide for visits to the historical sites.
If you were to visit Monreale from Western Sicily, you could drive an alternate route over the mountains that would avoid Palermo - Scopello to Monreale route question:
Or you could park at Palermo's airport and take the bus to Palermo and then the bus to Monreale from Palermo.
Where to stay in Western Sicily?: