Can't speak for Mykonos, but in Santorini, you must:
1. Go to Oia to see the sunset (but get there a few hours before the sunset to claim a good spot)
2. Sunbathe at Kamari Beach (absolutely beautiful)
3. Check out Red Beach (a bit of a hike so don't do it in flip flops and a beach dress like I did)
4. Stay at Stavros Villas in Karterados! It was cheap, beautiful, and Stavros invites everyone for coffee every morning and gives you these delicious little cookies to go with it. Also, see if there's a soccer game going on. The most fun I have ever had in my life.
Hope this gives you a few ideas! about 8 years ago
Imerovigli is a must-visit. Highest point on the island, best views. Hike up to ancient Thira (check the hours the actual site is open first, it closes earlier than one would think.) Great views from there though. Oia is a great little town, everyone says see the sunset there, I liked Imerovigli better. almost 6 years ago