I grew up in the Albany area. I've climbed 20+ peaks in the High Peaks area and my profile pic is of me snow shoeing in Lake Placid this past winter. There are a ton of great places to hike there, too many to even describe.
The Adirondacks are great and Lake Placid is a historic and wonderful thing to see..
The Catskills are great.
The Berkshires are great.
The White mountains are a bit farther and worth the drive.
The Green mountains are great.
I was in Schenectady this past weekend and wanted to do a fast paced exercise hike close to where my family lives so I Googled "hikes near Schenectady" and ended up doing a 7 mile loop at a place called Plotter Kill Preserve. It was beautiful
Join Meetup groups for hikers and campers and you will be booked solid for the rest of your nature lov'n life. http://www.meetup.com/