Mr Fly,
How long of a visit? I assume you're driving and bringing all of your own stuff, otherwise, options will be limited.
As Paul questioned above, several factors will affect any recommendations.
Solo or with your own group?
One vehicle or more than one?
Wilderness base camping?
Wilderness through tripping?
Public/private camping?
Some combination of public/private and wilderness?
Can you carry? Will you carry? Max carry distance?
Preferred water type,---pond hopping, lake travel, river trips, canoe bushwhacking...
I have been paddling and camping in the Adirondacks for a long time (no need to use harsh numbers), but I only ski in the Catskills (and only at Plattekill and some non disclosed back country spots), so I can't speak to Catskill paddling.
With that said, and without your responses, there are a few spots that a newcomer should consider not to miss.
Lake Lila and its environs. Minimal carry, lots to see and do.
Lows Lake, including Grass Pond and a scramble up Grass Pond Mountain. Minimal carry, more paddling than Lila.
Long Pond including Turtle and Slang Ponds (skip Hoel) Minimal carrying, fairly intimate and protected waters.
St Regis Canoe Wilderness Area and all of the included ponds. Carrying based on your ambitions and abilities.
Others here will have their can't miss lists too.
Based on your screen name, I assume you know the various bug seasons. Early season (soon after ice out sometime around late April early May) will miss the black flies, but some seasonal use roads will still be closed to the public.