Even if you don't hire a car in the Algarve there are many coach excursions available. These often feature the Monchique mountains. For those of you reading this who ski and know what a mountain looks like (You know - The Alps, Rockies etc.) then you should be warned that these 'mountains' of the Algarve are really foothills that lead nowhere. At the top, one can see some 'Alpine' like pasture, along with the mandatory, bell slung, cows, but that is all. More information can be found at www.monchique.com.
The area is worth a visit, especially in the hottest part of the year, as it is much cooler than the coastal fringe or the interior of the Algarve. The hills are well wooded, with eucalyptus appearing to be dominant. Monchique itself is a pretty place, famous for its spa water. If you feel like trying the Portuguese medical facilities then you might like to try a sample of the water (which tastes vile according to most people). The more hardy and robust can try the water at wayside springs rather than in the spa buildings.
If you do want to hire a car, and are brave enough, then you can tour through these mountains, stopping when you want. The following full day trip was successful in autumn 1998.
First fill the car with petrol then buy a good map that covers not just the Algarve but inland as well (if you can see the Barragem de Santa Clara on it - a huge reservoir - north of Albufeira, then it goes far enough). Do trace this route out before you start - you have been warned!
Head for the IP1 and go north along this "Motorway" towards Lisbon. Eventually near Sao Marcos da Serra you will find a small road to the railway station on the western side of the IP1 (Do note that some maps seem to have the station in the wrong place and also show a new road that was not there in 1998). In 2000 there is now a motorway style crossing so you turn right and then go over the motorway. Cross the railway track and head west towards Nave Redonda. This is a twisty road, full of repairs and some potholes, but keep following it and eventually you will come to a T junction with the EN266. Turn south (left) and now the "fun" starts as you climb a very twisty road up into the mountains. In 1998 this road was being re-surfaced and made the journey even harder. Eventually your determination will be rewarded with superb views across the forests. There are many parking places some by shrines built by the road constructors. After many miles of hard driving you reach Monchique itself and can drive through or stop as you like. After passing through the town (one way systems so be careful) you can detour to the main tourist spots or continue south towards the coast. Eventually you have another choice... to continue into Portimão or take the EN128 east to Silves (well worth a stop). From here you can head south through Lagoa to Carvoeiro and recover on the beach! Monchique, Silves and then Carvoeiro is a full day tour, but you can break the journey at many points.
Now, if you want more adventures then why not try a very similar trip in Australia. The trip from the Blue Mountains (near Sydney) to the Jenolan Caves is almost identical in the road quality, the scenery and the Eucalyptus trees!