I certainly would go to Elia Beach during your stay, for a change of scenery. It is quiter than the other "popular" beaches along the south coast like Paradise, Super Paradise, and Platos Yialos. Here there is plenty of room to swim, walk, and also shouldn't be a problem finding an umbrella and sun beds as it can be at some of the other beaches. The bay is bigger but still protected and it has a really cool laid back atmosphere. If you're standing with the sea in front of you, the part of the beach furthest to the left is private and belongs to the Royal Myconian Hotel, but the rest is open to the public. At the far right, at the very ends by the rocks, there are some nudists and gays, so if that's you, head towards the right! Otherwise, the middle is just fine, too! There are watersports here like at most beaches, and I think this is the best one for jet-skiing if you're into that. There are a couple of beach tavernas where you can eat--one has elevated seating (upstairs)--I don;t really recommend that one--not bad, but not as good as the one closer to the beach, next to the pool.
If it's water-skiing, para-sailing, windsurfing or boating that you like, then Elia Beach is the place for you. This is a lovely long sandy beach on the South Coast of Mykonos and there are plenty of tavernas to drop into for a cool drink and a bite to eat when you have finished. If you just want to relax in the sun, you can do that too as there are umbrellas and sun lounges provided and not as many people as at some of the other beaches.
Elia beach can be reached by bus from mykonos town in about half an hour, in 45 minutes by kaik from the village of Platis Yialos or in about 20 min by car or scooter. It's one of the best beaches of Mykonos, with a nudist part and Id say that the vast majority is gay... I personally found it a bit too much and probably didnt manage to enjoy properly the beauty of the place and the relaxed atmosphere because of too many naked male 'truck drivers' bodies - apologies if it sounds horrible..
There's a really lovely estaurant on the beach - on the right side and we had some delicious seafood and wine...