Rock Climbing in New York State consists of three main regions: the world-famous Gunks, properly called the Shawangunks; the Adirondacks, a vast state park with dozens of crags scattered across an area the size of Rhode Island; and Moss Island, a small crag near Little Falls. Several other areas exist, e.g. bouldering in Central Park, the Timp in Harriman Park, as well as many places of questionable access and/or worth.
With the rise of Sport Climbing, several places once illegal and/or dangerous to climb at now bristle with gleaming metal, providing another facet to NYS's menu. Many of these places still exist under tenuous legal conditions, so check with the locals and be discreet.
A few crags relegated to the dustbin of history have been revived. These include places like the "Powerlinez", a crag that saw occasional traffic up to the early 80s but then became off-limits and fell into obscurity; and Thacher State Park, which finally opened to legal climbing in 2017.
The main rock climbing venues are near the eastern side of the state, so folks out in Rochester, Buffalo, etc. have a tough time of it - they're either traveling a long ways, pumping plastic, sneaking into the Niagara gorge, or driving across the border to climb in Canada.
Ice climbing is a bit more evenly spread out. The Adirondacks holds the most reliable and extensive ice, but the Catskills generally forms a lot of good ice as well, and the Finger Lakes Region boasts a few lesser-known classic lines (many of which, alas, are not legal to ascend).
Alpine climbing is limited to the Adirondacks, though a few Catskill peaks come close. None of the mountains are particularly tall - Mt. Marcy, at 5,344' is the highest - but the weather is capricious and runs the extremes. Winter ascents of the High Peaks can be very serious endeavors, especially when climbing a classic line such as the Trap Dike or Gothics North Face.
Although a small state compared to the big West, travel in NY, especially to its mountainous regions, can be tricky. Public transport is decent within two hours of NYC along the major transport corridors, but off these or farther away, car rental, good maps, and good luck are required.
The great majority of visitors arrive in New York City; for those who do, the Gunks is a 2 hour bus ride to reach, and is definitely the place to go for rock climbing. The Catskills is a bit farther north, and having a car is almost mandatory, particularly during ice season. The Adirondacks is 4 to 5 hours away, and also requires a car. Moss Island lies along Interstate 90 and buses stop in the neighboring town of Little Falls.