You have to copyright the name, if nobody already owns it.
Most local governments have what they call a "fictitious name" registry. You can register a fictitious name, but that local registry doesn't stop someone else from starting a business with the same name in a different state.
A trademark is more of a national protection for a name. You will find that almost anything you can think of has already been claimed.
You may have to go to "open mic" nights and "Jam Night" at some local bars or clubs to be able to hear and be heard by local musicians. The benefit of that is that you get to hear them and they get to hear you.
You would need to know some standard songs so that you would know some songs in common.
Start going out to some places like this and just observe what's happening and see if you like how anybody is playing. If you feel up to it, get onstage and do some songs with the house band. Try it on for size. See what you do well and what you do not so well. Learn from the experience.
The more you do something like what is described above, the more you will learn about your local music prospects and the more idea you will have about whether it suits you or not. If not, go somewhere else and see if that's better. The more you get out and explore, the more you will learn about possibilities.
I got into the band I'm in now by answering an ad on craigslist. I've been playing two to three nights a week (while working full-time in the daytime) for the past six years. Some of the musicians you meet will be total flakes. Some will be your best friends for life. Be careful, but get out there and find your musicians. They are out there!
I've been playing in bands since before most of you were born. Lots and lots of gigs. It's a great thing to have in your life. Get started.