Register an LLC not DBA..
In most places DBA is basically a sole proprietorship, meaning no protection from personal liability. You are going to be a personal trainer, there is a lot that you can get sued for, LLC will cover you personally, so if they sue they sue LLC not you. Tax wise youre treated as a person, so no implications as with C-Corp
Once you get an LLC get a state and federal Tax ID. The federal Tax ID is called EIN# (Employer ID Number), don't let it confuse you, you need it for everything from taxes to bank accounts, even with no employees. A lot of states consider use of personal accounts for business purposes as methods to pierce the corporate veil, so make sure that EIN is tied to everything in the business, not your SSN.
As steve mentioned in the comment, insure like crazy, especially if you deal with clients in a personal setting, i.e. a home. Additionally, I'd suggest not doing anything in your own home, due to possibility of someone getting hurt and suinfg. Your best bet is partner with a small local gym or studio, and work out of there, the percentage you pay them will be priceless. It will not only give you a more professional appearance, but add client exposure, and lower your liability.
In NY, DBA's are handled by county clerks and cost about $40 which includes two notarized copies, one your bank will either keep or make a copy of when your open your business account. You can use either your personal SSN or a Tax ID number. If you use your SSN I believe you can add your Tax ID in the future. Call your local county clerks office they should have the information for you if not on their website
Not all states allow LLC's - so you could also consider an S-corp. All the answers about limiting liability should be first and foremost in your mind, especially given the nature of your business. DBAs do nothing to limit liability. I would also avoid using your home as much as possible.