A week in Sardinia - suggestions needed

We are a couple that will be going to Sardinia, and are looking for suggestions of accommodation on the value end and unusual things to see, preferably involving majestic nature (my thing), breath-taking architecture/cities (her thing) and seaside to die for (both).

We naturally have guidebooks a plenty, but are looking for actual inside suggestions more than the beaten track. Will be flying in/out of Olbia and will have a rental car for the entire trip, so mobility is not a problem but with just a week and mountain roads to navigate / things to see we’d rather not spend the entire time driving.

So, locals or fellow travellers – what should we not miss, where should we stay without paying a fortune and what would make this trip go that extra mile? Suggestions are much appreciated, even wild and crazy ones. We’re pretty flexible when it comes to itinerary.

Many thanks!