I've only been to Ajaccio a couple of times on day trips, interesting place tho the Bonaparte links tend to dominate. Corte, Propriano and Piana are stunning and doable from your base though the roads are challenging going north. One thing to note is that it's an incredibly mountainous island with about 50 of them over 2000m... this means that journeys can be long and hard in the summer with so many hairpins to tackle.... the bonus being that the views are simply the best I've ever seen. Bonifacio and Porto-Vecchio are stunning and the coast roads much easier to tackle.
Language is a little odd, mix of French and Genoese, local wines are excellent and the food outstanding. I'd hope to make it over there again this summer for a short break before the French arrive in August... I've used St Florent and Calvi (home to the French Foreign Legion) as bases previously and never get to as many places as hoped due to the challenging driving and terrified family on the mountain passes. Still my favourite place and still unspoilt by tourism. Enjoy and do try the Sanglier!