First off, if you go on a trail you should never walk without snowhoes or skis. It really pisses people off. We have alot of snow up here. You can rent a pull sled from High Peaks Ski shop in town. Quick hikes, Coppers & Winch Pond. Ponds are frozen and it's beautiful, between 1 - 2 miles right off 86 on way to whiteface. Cobble hill, 1-2 miles, steep the short way, easier the long way. Right off Mirror Lake Drive, Great views of town & HPKS. Lastly, trails behind Howard Johnsons. Goes out to Lake Placid with great views of whiteface.
Hope that helps. Have a great time!