You should play the nine holer at Pine Cliffs, which is really nice and visually spectacular. I personally think the best course on the coast is now Palmares, after the recent RTJ2 rebuild -- they've used some extra land to create three really nice loops of nine, and the holes down by the water still have the linksy feel they had in the original Pennink design.
Monte Rei is good in a big scale, tough Nicklaus kinda way, not really a course to walk though. I hear there are good deals to be had there, and frankly that probably applies most of the way across the Algarve. Don't be scared to ask for a discount.
Finally the very recently completed Beau Welling redesign of Quinta do Lago North is very good. Part of a long-established villa estate, so no way the routing could be changed, but I thought they did a very solid job.
Also, if you can -- and if it's still open -- try to eat dinner at the restaurant Vila Lisa in Mexilhoeira Grande, a little bit inland from Palmares, close to Penina (of which I'm not a great fan). I have never been but everything I have ever read paints it as the most authentic food experience to be had in the area -- the owner and chef is an artist who opens restricted hours to allow him time to paint, you eat what he cooks etc etc. Will need some research -- I am not even sure if it still exists, though it's a place I have wanted to go for ages. But food in the Algarve, and Portugal generally, is always good. I think Portugal has the best _ordinary_ food of anywhere I have ever been. It is quite hard to get a bad meal. I vividly remember the last lunch of a week with my wife and my parents spent in Alvor a few weeks after my brain surgery. It was October, and we sat at a beach bar eating grilled baby squid. They were superb, but we headed back to the airport covered in squid ink!
[edited to add] If you have a car, think about getting away from the coast for a day. Silves and Monchique are both worth a visit and if you don't mind a bit more driving, the Atlantic coast is beautiful. The village of Aljezur has a wonderful ruined Moorish castle, the Atlantic beaches are spectacular, though I guess it could be wild at this time of year, and there won't be many tourists around.