Choosing a studio depends a lot on what you want from your dancing. Are you interested in learning to social dance or do you want to compete? Can you afford private lessons or are you looking for group classes to start with? Are you looking for a social environment in the studio or do you prefer to just show up, take your lesson, and go home? Are you ready to commit to a fairly long-term process of taking lessons, or do you want to be able to test the waters a bit without much commitment?
Once you decide what you want, the best way to choose a studio is to check out a few and see which one you like best. Check out their websites and call them up to discuss what you are looking for. Most studios have introductory specials where you can have a lesson or two and often a group class for a pretty low price. If I were looking for a new studio, I'd take the special at a couple of them before I decided.
Opinions vary a lot about the chain schools, Arthur Murray and Fred Astaire. Some people love them, some have had less happy experiences. They often have a very good social environment. But you should be aware that they tend to be more expensive than independent studios and are geared toward selling you fairly long contracts for lessons, so you should consider if that's what you want before you sign up with them.
Welcome to ballroom dancing!