If you are prepared to start at 6.30am then I would certainly suggest Lisboa (Lisbon) as you drive through the Alentejo countryside north and if you have a good guide then you will also learn a lot of Portuguese history and other sociall facets. The capital city pffers a lot to see.

Sorry to disagree with Dan,but I think that a day trip to Lisboa from a seven night stay on the Algarve is not to be recommended because :
a)One day(or half a day by the time you allow for the return journey)gives you insufficient time to even get a flavour of the city, and
b)By the time you have had an early night to prepare for the long day ahead and then recovered from the journey on the following day,the best part of two days of your seven day holiday will have gone.
Go to the Algarve,fall in love with the people and their country;return again for a longer stay and then take time to see Lisboa or,better still,take a holiday dedicated specifically to Lisboa - it's deserving of an extended stay.
As far as Algarve excursions are concerned,could I suggest the Western Algarve trip for its stunning coastal scenery.The Monchique mountains trip is also scenic and interesting.If you can find a tour to the Alentejo region (often linked with a visit to a vineyard )you'll find gorgeous,uncommercialised countryside and excellent wines.
You don't say when you are going but,if your holiday is in the summer,you may very well find everything you need in the way of beaches close by.
You'll find that a week won't be enough on the Algarve - we usually go for three weeks and are never ready to retun home !

There is a lot to see! It so depends on your interests.

As per Sagres suggestion:
I would take a break from the beach scenario and visit Monchique.
And go for a guided nature walk!

I would make sure I sample some of the gorgeous food served in one of the better restaurants ! A great day out.

I like the Algarve mainly for it's fantastic beaches and coastal scenery though I hate to say it but the resort you are staying in is not one of the more attractive ones in my opinion. It's ok but spoilt a bit by some rather unattractive apartment block complexes along the sea front.

Apart from the great beaches and Golf courses, both of which are some of the best in Europe, it depends what you like to do on holiday. If you like theme parks, Zoomarine at Guia is pretty good though the rides are fairly tame and on the small side. It has a dolphinarium with regular dolphin displays, a Peter Pan-themed sea lion show, a large swimming pool area, a birds of prey display, the obligatory parrot show and a cinema where you can watch a short animated film in 4-D which is good fun. The Algarve also has two good waterparks, Slide & Splash and Aqualand - the big one (close to your resort), if you like that sort of thing.

A visit to the pleasant town of Monchique on the way to the top of the mountain at Foia is a popular trip though the mountain top itself is spoilt somewhat by numerous Aerial masts. If you like marinas, the one at Vilamoura is huge. Albufeira and Lagos are larger towns that have some character to them and are pleasant to wander round. Silves has a 11th Century Moorish castle and Cathedral, Tavira is an attractive town and Carvoeiro has retained much of it's fishing village character. You could also take a trip to Cape St. Vincent, the most westerly point in Europe with it's 19th Century lighthouse. None of these are what I would describe as must-do trips but they will give you a good feel for what the Algarve has to offer.

Just a quick OTT, I had the absolutley WORST meal ever in Monchique, it was in the cafe at the bottom of the steep hill on the right, the cafe with the iron ornaments!
However if not Lisbon then as Sagres suggest, Western Alarve. Cab Sao Vincente etc. You will also pass through SAGRES!!!!! well the village anyway!