If all you want is a week of rest with the odd night out then skip this section. You're probably staying in Maspalomas or Playa del Ingles and are planning to spend most of your time around the pool or on the beach.
On the way down from the airport all you saw was dry mountains and dust!? There probably isn't a part of the island more ugly, windy and arid. Here in the south almost everything was purpose-built for tourism from the 1960s onwards but the other 80% of the island is known as the "Continent in Miniature" and Unesco have decided most of the island should be protected as part of the World Biosphere Reserve and most of the island falls into the category of National Park.
As I've recommended elsewhere, if you can drive and are coming for a week then hire a car from the airport for the 7 days. An airport transfer or taxi is around 65 euros to the south and back and if you're lucky you should get car hire for just under 90 euros for a week.
So from the list below you've got info and photos on the more interesting towns and villages around the island, each worth a visit.