The owner of the studio where I take lessons often says that the hardest part of learning to dance is walking through the door the first time!
Check out a few studios in your area. Most will have websites so you can get a sense for what kind of classes and lessons they give. Call a couple that seem like what you want. Ask them about beginner groups and/or intro specials on private lessons (whichever you're interested in). If they are not friendly and helpful over the phone, move on to the next one. By the time you've called them all, you'll have made a connection at the one you like best and it will be easier to go in there.
Also, there will absolutely be other people in your same boat in any beginning class.
I assume you mean you're not physically flexible, as in touching your toes, etc. Don't worry. You don't need great flexibility to start dancing. I'm about the world's least naturally flexible person, and my teacher has made a dancer out of me! Plus, dancing is very motivating (addicting, even). Now I work on stretching and have gotten somewhat more flexible.
Welcome to dancing! You'll love it!