yes, in winter it is actually required to hike with snowshoes or crampons, but if youre not going so soon, there are TONS of great routes to hike.
step one would be to go to a camping store and pick up a trail map of the high peaks region, I plan all of my hikes off of the trail map.
what i like most about the high peaks region is that the mountains are actually pretty high, and have a tree line! so youre not just hiking in the woods all day, you get a change of terrain and scenery, and on clear days you can see incredibly far.
there are lots of great leanto's that you can hike to and use, i prefer the leanto to a tent...more roomy, and in some areas you can have ground fires, which is nice.
you can do alot in 4 days, the mountains are pretty close by so you can hit alot of peaks quickly, which is nice. if you dont want to rough it, you can stay at Johns Brook Lodge (when they are open) which is a nice little cabin about 4 miles in that is a great base camp for multiple day hikes with some of the higher peaks
heres some links to some photo albums of a few of my trips, you can get an idea for the scenery and terrain. I usually do my trips in late summer, but we did our last trip in october, which was great to see the colors on some leaves.
its a 5 hour drive from north jersey, and definetly worth it if you like actual mountains
http://www.facebook.com/album.php?aid=2033758&id=54900311&l=d1144
http://www.facebook.com/album.php?aid=2033764&id=54900311&l=bc79e
http://www.facebook.com/album.php?aid=2018986&id=54900311&l=e18ba