I know a few free and -VERY- basic domain sites. There's freewebs.com and I thinkg AngelFire is still open. Maybe you can try those first? By the looks of it, you'll probably need to pay money for a good domain site unless someone here could help.
Good domains don't come free. If they do there are strings attached with any professional would not like. Then comes the Subdomains but that too isn't a choice of a professionals.
Unless you are willing to have a really catchy name, the rates of domains aren't too high. You can get a regular domain name maximum within $ 50.