Iceland Air is pretty good. Kinda no frills. But it's a decent airline. you've got to pay to get a bag on board, but if you're good at traveling light, that's not a problem.
When in Iceland, you definitely want to hit the Blue Lagoon (you can do that in a short stop over, they have plenty of shuttles out and back), it's close to the airport and a cool experience. Everything in Iceland is expensive. You're just going to have to chalk that up and deal with it. If you're there for a few days, you can load up in the duty free on booze, supplies, whatever, and it's generally cheaper than buying in Iceland itself. Golden Circle is worth the trip, a number of the different tours are worth your time. I loved getting to the Black Sand Beaches, but if you only have a day, it might not be possible. If you're going in the summer, you can just rent a car and hit up the sites. They're typically privately owned, but Iceland has a pretty firm belief that even though a family might own the land that a waterfall is on, you should have access to see it/go to it. So you can go see a ton of stuff without paying, just as long as you have a GPS/map (it's not hard), and are respectful of the sites. Just expect to have them want you to buy some stuff (usually a coffee or a coke when you hit a site is pretty common).
The people are great, it's a ton of fun.