The state of New York actually has many resources for those looking to start a business. You can find some of them here:
Meet With a Small Business Counselor In New York State. You can even schedule an appointment for free business counseling.
A DBA in New York is called a Certificate of Assumed name. You can find that information here:
Corporations Frequently Asked Questions (it's under the Brochures section).
No, t-shirt companies are not exempt from taxes. Generally, the only organizations exempt from taxes are non-profits.
You probably also want to read this pamphlet here -
http://opal.gorr.state.ny.us/gorr/pa...e2009.pdf?Open
Keep in mind that you should pay self-employment tax. Self-Employment Tax is a social security and Medicare tax for individuals who work for themselves. Social security benefits are available to self-employed persons just as they are to wage earners. Your payments of self-employment tax contribute to your coverage under the social security system. Publication 533, Self-Employment Tax, may be obtained by contacting the Internal Revenue Service at 800-829-1040 or at their website:
Internal Revenue Service.
Finally, here is a great website that will give you the overview of what you'll need to do:
Starting a Business in New York | Business.gov
need some bussiness guidence (NY) 
