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(do)I just claim the additional income on my own tax stuff. then do all the deduction for like supplies and crap the next year?
In a word, "yes". There is no need to do anything more than keep records of your expenses for tax deduction purposes. If the business takes off to the point where you need to hire people and rent space for it, then you should consider Setting up an LLC.
You Would set up an account with your State's retail sales tax division. It would be in your name doing business as, i. e. D/B/A (make something up). Anyone can be a business in their own name.