I have the state tax # (got that first), which was free...you just fill out the forms and mail it in and they send you a resale certificate. I have to file and pay the sales tax annually.
Locally, I got a DBA (doing business as) form, for which I had to go to the county offices, and do a search to make sure there are no other businesses with the same name. I had the form notarized (a few bucks) and I think the filing fee was $35, one time fee. If you don't get the sales tax #, the county will report you to the state, so that's a must, but they tell you that when you file here in NY.
You need the DBA to get a business bank account, if you want to operate without using your own name as the business name, or want your business name printed on your checks, etc.