If you do a search on the forum Im pretty sure Ive given advice a few times on this question.
I go there 2 or 3 times a year (my wife is from Corsica)
If I only had one holiday left in my life Corsica is where I would chose to go.
Great mountains, brilliant forests and lakes, foot hills full of little villages and olive groves. Amazing coast with incredible beaches. Great food and drink, real culture. Good things to do like cannoning, biking, diving, climbing, rambling, sailing, walking etc.
Hire a car and go and explore, anywhere will do, its all great and pretty small. You can be on top of the mountain for breakfast, lunch at the beach and eveing in a little village high up with a view.
Don’t go in August – too hot and too full of French / Italian tourists. Best time is late Sept.
Email me if you want more specific info. Oh, and there is some mountain biking too.