Hi
My favourite beach in Algarve is "Senhora da Rocha", a cove near "Armação de Pera", 50kms from Lagos. I'm not the kind of person who likes to stay laying in the sand, so I need something to do while in beach, and this one has a lot of funny things: fishermen coming and going, a restaurant in the beach (at 11, more or less, they start picking the fish they will serve from the nets that are in the sea), a underground path (under the cliff) that leads to another beach, etc. Parking there is not easy, but it is possible. There are boats to take you to see hidden beaches and grottes, like "Benagil" (search google to see how beatiful it is). You don't book this boats in advance, just go there and ask the fishermen what time it is possible to go.
If you want something more touristic, at Lagos (or anyother city in Algarve) you'll find a lot of advertisements for "Pirat" boats, etc, that will take along the coast with animation, and will serve you lunch in a desert beach along the way.
Near Lagos, just across the bridge (way to Portimão) there is a nice, long, golden sand beach: "Meia Praia". Very similar ot this one you have "Luz", on the way from Lagos to Sagres.
Helena