I go there between once or three times a year and have been for the last 10 years.
Basically its a pretty small place, so if you hire a car it wont matter where you stay.
However I would suggest the West Coast. Then just explore the island. The East coast has a long flat bit which is not as exciting. But everything is so near your either 40 mins from the top of the mountain to the beach.
For beautiful mountains pick Zonza, or stunning seaside up in the North West say Porto or in the South West Propriano. The capital is OK, but its a busy town. Whereas places like Bonefacio or Porto Vechio are smaller towns and full of great places to eat and stay.
Id not bother going to Sardinia if your are in Corsica. Corsica is similar but has more forests as in Sardinia they have cut most of them down.
Dont go in August, its too hot and too busy. Late Sept is good, or even later. Or perhaps May / June.
(Oh, and in Zonza there are some good mountain bike trails)
My favourite village is either Zonza or Sartene. My favourite beach is Campomoro and my favourite town is Bonnefacio.
Try this link for accommodation, its French: http://www.leboncoin.fr