Sorry I haven't been back sooner. I've been hoping Cubsur would be along to tell you how easy it is to get to various places by public transport. I'll give him a prod!
The Old Town has some very interesting old buildings and townscape, although the main square and some of the esplanade has now been "modernised", so I think you will enjoy exploring that area and sampling some of the excellent cafes and restaurants. For an overview of some of the cultural sights and sites you could take a look at
this website. The towns you might like to visit would be Faro (Old Town) and Tavira to the east and Silves and Lagos to the west.
I would imagine it would be possible to get to all these places by public transport but we really need Cubsur to help on this one. Another alternative would be to use a taxi firm who would take you on a day trip tailored to where you wanted to go or suggest one based on your particular interests. I haven't got any experience of using these, but I've read good reports of Alan Sutton of Owners in Portugal. Read this
TA forum post and I think you'll agree that it could be for you.
The beaches are superb all along the coast, ranging from the very long Praia da Falesia with its red cliffs, through the coves and beaches of Albufeira and the typically Algarvian coves starting just beyond the Marina to the long stretch of sand starting at Gale and ending at Armacao de Pera. To see all these have a look
here. You can walk along the cliffs or on the beaches, but since the cliff fall tragedy at Maria Luisa we're all a bit more aware of the dangers of walking too close to the edge of the cliffs.