From the rest area parking lot on the south side of Vail Pass, look for a trailhead and dirt singletrack just before you get the to paved bike path (on the south end of the lot). The path quickly crosses a creek (with no bridge, plan on getting wet in "early" season, which can be July at this elevation!).
Go across grassy terrain that is mostly level for a while, passing a marked intersection that has a connector to the paved bike path, and then turn up Wilder Gulch proper. Follow a stream past many beaver dams, mostly through wide open meadow. The flowers can be spectacular in July and August. There are many small side-streams to cross, some with rock armoring, but you may have to dismount a few times.
Thomas Wilder is rumored to have lived here in the 1880's and cabin remains can be found if you really hunt (about halfway up, just past the last beaver pond).
Keep climbing with views of Ptarmigan Peak
and Ptarmigan Pass as the trail gets wider and becomes an old two-track. You'll arrive at a large flat meadow that makes a good destination for a short ride. Continuing, you'll start to climb more steeply through woods before ending at Road 747.
It is highly recommended to take a left on Road 747 up to Ptarmigan Pass, where the views are spectacular.
This entire area is a very fragile ecosystem, and it's popular with hikers, so be on your best biking behavior: don't go off trail and be extra courteous to everyone you meet.