No clue about the NY stuff... but...
I've been shooting since I was 4 and got a shotgun for my 7th birthday. Gun ownership and hunting has definitely fun and rewarding.
Gun safety classes are easy to find. The #1 rule is to ALWAYS treat the gun as if it is loaded. ALYAYS! Never have it pointing at anything you could not tolerate receiving some instant ventilation. This one rule can avert most any accident.
I know some gun clubs have indoor or outdoor shooting ranges and set up various scenarios. However, I'm not sure about finding a home defense "course". Gun club members are typically people who have been shooting for years but I'm sure they would be happy to help you. Just check around at a local gun shop for a gun club.