To be totally honest, December is mid-winter here in the Algarve and many activities of interest to teenagers will not be operating.
However, some ideas (Most of these will require a car):
1) History:
The extreme south-west of the Algarve region of Portugal/Iberian Peninsula was critical to the Portuguese Voyages of discovery that opened up the route around the Cape for Europeans. You should visit the Fort at Sagres built by Prince Henry the Navigator as well as the lighthouse at nearby Cape St. Vincent, the most South Westerly point in Europe.
In Praia da Luz, itself, there are still some small ruins visible of a villa dating back to Roman Times.
8km from Praia da Luz is the town of Lagos where I live. In my opinion it is one of the few towns in the Algarve that retains its Portuguese ambiance and has not yet been taken over totally by tourism, It is a historic city. Recent archaeological findings suggest the the town was settled by the Phoenicians.
See the following entry in wikipedia:
http://en.wikipedia.org/wiki/Lagos,_Portugal
Another historic place to visit is Silves, which was a key center during the period of Moorsih rule.
See the following entry in wikipedia:
http://en.wikipedia.org/wiki/Silves
2) Geographic
1) Explore the beaches up the West Coast of the Algarve between Vila de Bispo/Sagres and Aljezur. A spectacular coastline with hidden beaches and many that are world-renowned surfing beaches.
2) Visit Monchique mountain, the town of Monchique and Caldas de Monchique. This dominates the region geographically and there are great views.
3) Entertainment
For something different and if they are open...(their site is not very clear.) Zoomarine about 40kms from Luz maybe worth a visit, They offer interactive sessions with Dolphins.
See: http://www.zoomarine.pt
If you want to go further afield, Tavira, Castro Marim and Alcoutim in the East Algarve, about 75-120 minutes drive from Luz may be worth visiting from a historic point of view.
