The Algarve is a great winter sun destination for walkers. The sunlight glinting off the sparkling sea is a great antidote to overcast northern Europe. Winter is ideal walking weather - bright and sunny, but not too hot (shorts in the day, but sweater at night), and no problem with sunburn
Cost walks are the big draw, especially the XXX walk which must rate at one of the world's best walks - a cliff walk past kast (limestone) towers, arches, stack, towers and blowholes. You can follow the cliff path, or clamber through rock arches from bay to bay. This really is a spectacular walk. Think the island in the Jame Bond film, Man with the Golden Gun.
There are several other great coast walks around Algarve's south west corner (i.e. west of Lagos, and the west facing Atlantic coast).
There are a few inland mountain walks
In the winter, £200 for a 2 week package holiday (or Expedia style flights plus hotel deal), and £50 for a week's car hire is possible. Most towns will be quiet outside of Christmas and New Year.
In practise, you will need a car to get to most walks. Outside of city centres, there is very little traffic. The Algarve motorway is now a toll road. There are no toll booths - you get seen by cameras, then pay at a post office afterwards. Which means you cannot use the motorway to drive to the airport at the end of your holiday!
Lagos (about 50 miles west of the airport) is a good base.
The Cicerone book is good, however, there is no travel information other than walking, and its hard to tell which are the OK walks, and which are the truly great ones.
The Sunflower and Rother are new editions since we were last there. Its useful to have a Lonely Planet or Rough Guide if you have any interest other than walking.
By the way, the photos were taken in January.
Ideas for Short Weekend Trips away from London, and Walking holidays in Europe