I'm a local, so a slightly different perspective, but I will give it a go.
Transport
A bike is useful depending on where you are staying, the Hampton is sort of pigeon-holed in a weird spot where a bike isn't very good. Please don't ride a bike on HW 98, please. Depending on time of year, traffic will verge from non-existent to very heavy on 98. In June... it'll be busy. Uber has worked for me a few times, but can't comment on using it as a main transport. Parking will be difficult at the public beach accesses, so walk or bike to those when possible.
Things to Do
Difficult to answer, depends on what you like. General watersports are always top of the list for most (jetski, pontoon boat, parasail, etc) send me a PM and I can tell you who to stay away from on that front. There are some art galleries if that is your thing. A few small breweries. Outlet mall. Fishing trips. Some very nice state parks and hiking areas. Harborwalk is nice to check out and if that is your scene you can stay there all day. I would go for a drive down 30A and see Seaside (Truman Show) and the state parks and forests there.
Restaurants
Again depends on what you like and your price point, some personal favorites are Harbor Docks, Boshamps, McGuires, and Dewey Destins (bayside, not harbor) for Destin proper. Back Porch right near that hotel is decent and gets a great sunset, but not top of my list for food. Just try to stay away from the chains, please.
Can't comment on the FBO, other than to say gas is typically kind of expensive - look at CEW, could be as much as a dollar a gallon cheaper.