Madeira is a really a beautiful island, but definitely not what are you looking for if you want to avoid Brits and Germans ;)
We went there at the end of September, and it was pretty windy, and a bit cold - I went swimming just 1-2 times during our two week vacation.
We went around the island (well, half of it), and the place that I really enjoyed was Porto Moniz, where we stayed about 7 days instead of the one we planned to ;)
Loved it - a lot of wind, rocks, magnificent sunsets and great dining...Let me see, there are some photos here:
http://www.flickr.com/photos/alegria275/se...57603923973481/The only beach on the island I heard of was Calleta beach, which turned out to be rather disappointing. There is however a beautiful beach island near Madeira, called Porto Santo where Columbus lived for some time. The only disadvantage is that it costed about 100 eur/person to get there from Funchal, and the hotels were pretty expensive (e.g. 200eur+ /night).
The island features magnificent tracking paths, wonderful nature - we've seen eagles near Porto Moniz, and there are absolutely beautiful flowers everywhere, and awesome seafood. Porto Moniz should be great in July, but if you're looking for the beaches and the but beautiful nature too, I can definitely recommend the Adriatic coast - e.g. the island of Brach.
If you decide to go to Madeira, just send me a PM, and I'll give you some more tips :)