The Catskill region includes Catskill Park, which is nationally unique because it is a checkerboard of public and private land. Hiking opportunities abound, with approximately 300 miles of marked, maintained hiking trails on public Forest Preserve land.
Stewardship and development of these trails is shared by the New York State Forest Rangers and local chapters of hiking clubs such as the New York/New Jersey Trail Conference and the Adirondack Mountain Club. There are 35 Catskill peaks boasting heights greater than 3500ft elevation, and people who hike to the summits of each of these peaks earn membership in the Catskill 3500 Club, an organization founded in 1962, to foster interest in hiking in the Catskills.
Kaaterskill Falls is the highest cascading waterfall in New York State -- a 260 foot, two-tiered waterfall that has attracted tourists, poets and painters since the early 1800s.
Whatever trail you choose, you'll be sure to enjoy the natural landscape, peaceful forests and mountain scenery.
For more information on the Park and hiking and other outdoor activites visit the NYS Department of Environmental Conservation.
For an informal list of some of the best hikes available in the region, check out Catskill Mountaineer, an organization dedicated to helping people learn about and enjoy the Catskill Mountains. Backpacker.com is also a good resource for official hikes as well as user-submitted hikes, including some on newly opened up private land. Trailkeeper.org has great listings for hikes on public land; plus maps, tips, and even some restaurant guides for a little post-hiking R&R.
Check out this story from the New York Times about the area around Big Indian and, if you're looking for kid-friendly trails read the Appalachian Mountain Club's Catskills blog.
And, when in doubt... check it out. Read up on the basics about hiking.