I used to live near there... Agreed with Eric - the closest place is Little Falls (Moss Island), which is basically like an outdoor gym and hosts a short 30' wall that is usually TR'ed. A little further is the Gunks, which is way better of course. If you are going to be near Rochester, RIT has a really neat bouldering gym built in a barn (The Red Barn). Good luck!
What kind of climbing do you want to do, and more importantly are you willing to drive?
If you want TRing, go to Moss Island in Little Falls. Climbs from 5.1-5.13, and lots of variety in movement/style. If you'd like to get on some very eastern (i.e. thin and dead vertical) sport routes, check out the many other crags around Little Falls (Dihedrals, Recycled Rock, or many of the newer developed areas, which are sometimes kept secret). You can also go to numerous areas just south of the NY border in PA for trad/sport.
If you want to do some quality trad lines, than the Gunks,Dacks, or Dihedrals are your best options, as well as some gorgeous cracks at Lost T Cliff.
Snowy Mtn. has a long hike in, but the climbing and scenerary is fabulous for sport.
Bouldering gives you two decent destinations: Niagara Glen if you have a passport, and Nine Corners in the Southern Dacks. Nine Corners is a gem of a bouldering area.
If you're looking for ice, there is a lot in the Finger Lakes, but you'll have to know someone for most of it ;)