And once you're done mastering NY Penal Law sections 265 and 400 - make sure you become very well acquainted with New York Penal Law section 35 (Defense of Justification) before you even think about carrying.
Article 35 - Penal Law - Defense of Justification
As for where you can't carry... generally speaking...
Court houses, municipal buildings, county/state/federal government buildings, school buildings and grounds, college campuses, any place that says "no firearms", state parks, airport terminal buildings (especially the part beyond the TSA screening area), etc.
Car laws - NY is a carry concealed state, so carrying in a car is considered concealed so you're kosher there. Transport laws - there are none if your permit is unrestricted / full carry. If your permit is restricted - there may be administrative restrictions on how you may transport your weapon (but these are NOT laws... just guidelines to follow lest you piss the judge off). For NYC - you have to lock it up in the trunk if you're driving through the city with NO stops until you're outside of the 5 boroughs.
Discharge laws - generally vary from town to town... some prohibit discharge within the town's boundaries... usually find these ordinances in more urbanized and suburban areas than rural areas.