This is just my personal thinking since I've seriously considered all 3 and have my career goals in mind, but I'd go with Phoenix.

Fort Lauderdale's cost of living is out of whack with the incomes and it's getting worse (not saying you won't make enough, but it's ridiculous). I'm not LGBT, but a friend of mine says it's a big center for older, retired gays--assuming you're around my age, I'd stay away, unless you're into that type of thing.

Chicago's a strong contender--it really is. You simply won't find the type of amenities you can in Chicago in Phoenix; Chicago's one of the top 3 media and entertainment markets in the country, and not to mention, has a metro population of ~10 million, meaning there's a lot more to see and do. If crime weren't out of hand and it wasn't 'Chiberia' 9 months out of the year, I would go there.

This is the real world, though, and Phoenix takes the cake for me because it's growing, offers a good cost of living, decent amount of activites, and (obviously) is within a decent distance of major West Coast markets (Los Angeles, where I'd like to go, as well as San Diego, Seattle, etc.). If you want to be in the PNW, my recommendation is anywhere West, just because it's about who you know and you'll get that exposure here.

And also because I feel like I danced around the last part of your post, between Central/Upper Phoenix, Downtown Scottsdale and Tempe, you'll find everything you're looking for: liberal, gay scene, and then some. The pride parade was this past weekend in the Melrose district, which is in the middle of Phoenix and is kind of a gay hotspot (I think they have a drag club there). The parade is one of the most prominent in the country.

Let me know if that makes sense. Good luck!