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That's very little time, given how much there's to see and do on Corsica - no time for a hike in the mountains, I'm afraid.

Bastia is the least interesting of these towns (though I did like it - but I was in Corsica for 2 weeks ) and a bit out of the way, so I'd skip it.

The train ride from Ajaccio to Corte and on to Calvi is highly recommended (I suppose you'll be travelling by public transport). Each town has enough to see and do to keep you amused for at least one full day, so you'll need to be selective about the sights you visit). Their town centers are all quite attractive, but don't take more than an hour to stroll through. In Ajaccio, Napoleon's house is also worth a visit, ditto for the citadels of a Corte (which also houses and interesting museum) and Calvi.

In the end all depends on your personal interests, but what you could do is head to Napoleon's house upon arrival in Ajaccio, then take a train to Corte around noon (check the CFC website, because trains are fairly infrequent). Spend the night in Corte (we stayed at the Hotel du Nord on the main street of the od town, nice place; restaurant U Museu is highly recommended if you want to try some excellent local dishes) and take a morning train to Calvi. If the weather is nice, spend some time at one of the beaches in the area, or you could even take a boat trip to the Scandola nature reserve in the afternoon, if they are running at that time of the year.

This would be a pretty rushed journey though, better add a few more days if you can.

P.S. thanks for the PM

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Thanks again!