It seems you need a license to teach school (of course) and a license to be a handyman in NY. But you may not need a license specifically to teach handyman stuff. However, recognize that if you are teaching things like electrician work, you still might.
Ask at your local library or community center or Y. Any of those places might be a good place to advertise your services when you're ready to, so you may as well develop the relationship now.
You may also want to talk to your insurance agent or a lawyer before you continue. If you are teaching people how to run a circular saw or wire a lamp, either of those could result in injury if done improperly. Find out how to protect yourself if you get sued. Also talk to that lawyer about incorporating, probably as an LLC. That is, if you're ever sued, you want to try to keep your house.