Opposite side of Walden, but State Forest State Park has some great trails, good fishing, and good chances for seeing deer and moose. Thinking of the little ones, there are lots of trails that are less than two miles...Ranger Lakes Nature Trail, Lake Agnes Trail, Mountain View Nature Trail...and more. Mountain View starts right across from "Moose Overlook" which is basically just a platform overlooking some good moose habitat...I haven't seen any there but that doesn't mean you won't. All three of those trails are short, flat, maintained, and some have a bench or two along the way.
While you're in Steamboat you could stop by Fish Creek Falls...the trail and bridge closes seasonally but I think you can pretty much always get close enough to at least see the waterfall.