Welcome to NY and to VFTT, Casey. Albany's my hometown, and I always liked that it was sort of central to a lot of NY State and the Northeast. You're easily within dayhike range of the Adirondacks, the Catskills, the Shawangunks, Vermont, and Massachusettes, and a day's drive or less from New Hampshire and Maine. A couple places for nice easy walks right near Albany are John Boyd Thatcher Park and Five Rivers Preserve.
As RDL points out, you came to the right place w/ this board. As you learn more about the area and make your questions more specific, you'll have almost limitless outdoors options. There's some other good websites listed in the Links section. The NY State Department of Environmental Conservation (NYSDEC) is even a source of good, if very limited, information.
If you're looking for a place to buy maps and trail books, try the Book House and EMS, both in Albany's Stuvesant Plaza. (I have no financial connection or interest in either store, etc., etc.) Enjoy!
Matt