I've had a good think about this and there are loads of things to see, the only place we didn't enjoy was Cagliari. The beaches are excellent, the interior is spectacular and the public transport is good, trains and buses. We were towing a boat so went to places with jetties which were all good and interesting.
One thing I'd definitely do is take the ferry from Santa Teresa Gallura to Bonifacio*, birthplace of Napoleon. You'll pass the Maddalena Archipelago which is a fabulous group of islands and it's only an hour to Corsica. The arrival, into Bonifacio, is incredible and the town is interesting with lots of good restaurants ... it's French but the locals don't see themselves that way and have their own language. It's definitely worth a day trip, in my opinion, and a bit of an adventure. Just go as foot passengers, it's cheaper and you only have a nice walk into town.
* Bonifacio
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I'll post a few more suggestions later.