Is there a minimum amount of money to become taxable?

Is there such thing as not making enough money to get taxed in NY state? What is that amount?

In general, the answer is: yes, it is possible to make so little money that you don't even have to file taxes. Even if you don't end up owing any taxes, you may want to file tax returns anyway, for a number of reasons: to get back money that was withheld from your paychecks, to get the Earned Income Credit, or even just to eliminate the risk that the IRS will come after you and ask you to prove that you didn't have to file.

The answer to your question is probably somewhere in IRS publication 501. The rules are pretty complicated, so start by reading that.

As a student, you should consider if you are being considered a dependant on someone else's tax return (maybe your parents'). That will change things, too.

There are multiple type of taxes, some use a bracket system where taxes vary based on income level.

There are both federal and state income taxes.
Corporate income taxes, which dont really apply to an S-corp because the taxes pass right on to the owner. There are also Medicare, social security and unemployment taxes.

If you describe your business in general some of us may be able to give you some advice.