Hi Phoebe... I'm kinda new here also. I've flown all over the world (on sim) and definitely have some favorite airports.
KERV Kerville, Texas is the first that came to mind when I saw your topic. Its just a neat little airport that I really liked when I landed there for the virtual night on my tour of the US States.
KVEL Vernal, Utah is my home airport in the real world. I've sometimes been disappointed with how FS9 shows some places, but I have to say they got our own airport NAILED. I've flown in and out in real life a few times and now on the sim I get to taxi about on the very tarmac where I've stood to watch airshows, then out to the end of RW16 where I've watch many real aircraft start their take off's. They also did well with the scenery and the feel of our town and valley.
Salt Lake City, Utah is another where the sim got it really good. The mountains in the area are quite accurate on FS9 are are really fun to fly about.
I agree with others that have mentioned Honolulu, Tampa Bay, St. Marten (any in the Carribean are great... I especially liked St. Barts).
Hong Kong has a couple of fun airports. The new one is one of the busiest I encountered in my around the world trip. (Of course I didn't always have the traffic setting up real high). Taxiing the Vega about and trying to keep out of the way of a bunch of 747's was way fun.
But the Old Hong Kong airport is a great one to try landing a 747 at... approach from the north and land on RW 13... its exciting.
I also found some fun airports that present some interesting flying challenges. I just this morning posted a topic in this forum for one such airport. https://forum.flyawaysimulation.com/forum/topic/21354/flying-challenge-for-anyone-interested/
I found many neat airports on my trip around the world and could go on and on. If you still want further insights to some of those after you've checked out those in the replies here, just let me know.