We spent our honeymoon in Vilamoura, and had a fantastic time. We stayed in the Marinotel and loved it. As Lummie said, go DIY and you will save a fortune. We use
http://www.medhotels.com and
http://www.alpharooms.com for our hotel accommodation.
Vilamoura is a lovely chic resort, with a beautiful beach and a huge marina. Lots of bars & restaurants. All the hotels in the area that we saw looked to be of a very good standard.
Lagos is also a lovely resort, although we haven't actually stayed there, but we did spend a nice afternoon there, and we hope to holiday there some time soon.
This year we are going to Carvoeiro. Again we spent an afternoon there before, and loved the feel of the place. It's a small fishing village, and is charming. For the size of the resort, it has a huge selection of restaurants. It's a good base for seeing the rest of the Algarve too.
Praia du Luz is a small resort, with a fab beach. I've read some lovely reviews on the area, so it might be an option for you.
The best thing to do is look at the resort report section of this site. It's great for honest reviews, and will help you narrow down your choice.
Enjoy