There are many activities, waterparks, jeep safaris, boat tours, and so on. You will find leaflets and selling points everywhere, be sure to pick up the leaflets as they will usually give an immediate discount.
Here are some options:
Paderne Castle is about a 1 hour walk from the centre of Paderne as it gives spectacular views. So it's worth while on a not too hot, and preferably dry, day. You can get there by car too (but not with a Ferrari, as not all roads are paved). From Padernem just follow the signs “Castelo”. It's not more than a ruin really, but it is 9 centuries old and one of the castles on the Portuguese flag due to its significance in old times. The village of Paderne was actually situated near to the castle until the 1755 earthquake demolished most of the houses and it was replaced a few kilometers inwards where it is today. Arriving at the castle, watch how the river runs around the mountain on which the castle is built and you will understand its location. You can also see the medieval bridge in the distance (that some call the Roman bridge but there is no proof of that...)
Albufeira old town centre has street artists every evening to entertain the tourists, and many bars have live music. Some streets will feel like an open air discotheque but the atmosphere remains friendly at all times. Parking here is also easier than in the new centre (around the Strip). A pretty nice development is the new parking garage behind the townhall (just continue until the waterfront) where open-air escalators bring you downtown and back up again.
Albufeira has a bullfighting ring. Many people will disapprove of this form of entertainment, but it is part of the folklore in the Iberic peninsula. If you want to visit it, see the announcements in the town; usually twice a week there is a show, costs will be something around 30 Euros per person.
Portuguese bullfighting differs from Spanish bullfighting in some ways. For example, the Portuguese Cavaleiro fights the bull while from horseback, which makes it very spectacular. There are much more female bullfighters in Spain and the most spectacular difference is the “pega” . The pega happens after the bull has been fought by the Cavaleiro and is tired. Then 8 unarmed men step into the arena and challenge the bull to attack them. When it does, they immobilize the bull (which weights about 450 kgs) with their bare hands. This is impressive to see and for sure the most dangerous part of the show (the danger is entirely for the men and not for the bull). Another big difference: the bull is not killed in the ring (but don't be mistaken, it will not survive the spectacle).
ZooMarine is the biggest of the many local waterparks and has significantly more to offer than just water. It's on the EN125 direction to Portimao, some 4 kilometers after passing Algarveshopping.
Silves is a beautiful town with a pretty well conserved castle; the small medieval streets wander upwards to it and make for a nice day out on the town.
The marina of Vilamoura is worth a visit just for a walk or for a fancy dinner, good idea for a warm evening. It's easy to spot some very expensive yachts and cars. If you walk until the end, you will find many offers for boat tours, fishing trips, even private boat rentals, and so on. Vilamoura also has one of the few casinos in the Algarve (the other one being in Portimão as far as we know).
Fiesa is an impressive exhibition of sand sculptures, about 20km drive. Every year, there is a new theme and new sculptures. Very well signed, it is in the neighbourhood of Pera.
Second half of August there is a 10-day fair in Lagoa: Called Fatacil. Concerts, food and all kind of merchandise, from 6pm until 2am. Entrance fee is only a few Euros per person. Just follow the EN125 westbound and just before entering Lagoa it is on your left hand side.
Lagos has a beautiful Zoo, not too big but really well laid out with especially nice monkey habitats. The location is a bit off the beaten track, so it makes for a nice drive through a more rural area (good roads though, do not worry about that).
Alte is a beautiful old village in the mountains only 12 kilometers from Paderne. Drive north direction towards Lisboa, on the final roundabout, take a right to “Alte” and you'll easily find it. The view from the terrace of the Alte Hotel is among the best you can find in the Algarve. Not to miss is the area around the spring (“fontes”) which is well developed and has an amphitheatre and a bar/restaurant. Just follow the signs in the village.
Ferragudo is a still relatively small fishermen's village next to Portimão on the other side of the river with nice restaurants and a great atmosphere. And you can take the ferry across the river to Portimão...
Monchique is a village with very well known natural springs and beautifully laid out. While you are there, drive on to “Foia”, the highest point in the Algarve (900 m). On a clear day (of which there are many) the view is spectacular. The top has a coffee-bar with snacks and a shop with a nice collection of souvenirs, locally produced drinks and foods, and traditional clothing.
Cabo São Vicente (about 85 km): is the most South-western point of Europe with high cliffs, barren grounds and one of the most powerful lighthouses on the continent. Sweeping views over the Atlantic Ocean and fishermen doing maneuvres that we think are pretty scary, but they seem to find it normal. The city of Sagres nearby is cute and cosy (though the fortress isn't worth the money in our opinion). In fact, the whole area around there is a natural parque and it shows.
A nice daytrip is to drive up to Monchique, continue to the high point of Foia, then drive west to Aljezur, drive down along the coast to Cabo São Vicente, and drive back east towards Albufeira/Paderne. All together “only” some 220 km but make a day out of it and you will see a lot of the Algarve.
Tavira, easy to reach and a wonderful drive if you pass Faro to the north on the local roads. Is an old and nice city, with a the ferry to the Armona-island as a beautiful bonus. The camera obscura in the old water tower is very interesting too!
Armona-island (“Ilha de Armona”) is a stretch of sand with a few houses and some bars and restaurants that can only be reached using the ferry. The beach life there has a different feel than anywhere else as there are no cars. It's really summer style and worthwile for a different day-to-the-beach. The ferry is not expensive and in summer goes frequently.
Maybe the most simple suggestion of them all, but a proven success: follow one of the signs to a beach you don't know (regardless if it's summer or not) and you will most probably find yet another beautiful area of nature with great photo opportunities and maybe even an unexpected restaurant for a simple breakfast, a good lunch, or a drink while the sun goes down.