It is recommended to ride this route clockwise (due to several steep sections that are better ridden on the descent).
From the trailhead, pick up the Mountain Lion trail (starts at the restrooms). Climb until you come to an intersection with the Burro trail, and continue on Mountain Lion. You'll eventually pass a pond and an abandoned settlement on your left; keep riding and the fun begins shortly after. The trail will continue to climb, with sections of loose rock/sandy/rocky terrain. If you have to stop, due to an obstacle you can't clear or to take a breather, good luck starting again. After a few switchbacks, you'll get to the summit. Continue to follow the Mountain Lion trail. When you come to the next intersection (again with the Burro trail), be sure to take a left and continue on Mountain Lion. Descend a few switchbacks, some of which are quite rocky. You'll eventually drop down to Deer Creek and ride along the creek for approx. one mile; this mile has many rocky/technical sections and crosses the creek several times on old logs. You'll eventually come to another intersection (with a trail heading off the left to an old quarry); again, stick to Mountain Lion, and continue the descent back to the trail head.
There are several other trails that intersect with the Mountain Lion trail, some of which are for hikers only (no bikes). All intersections are clearly marked with maps, so pay attention and you'll find your way. There are maps on the state park website (http://www.parks.state.co.us/parks/goldengatecanyon/Pages/GoldenGateStatePark.aspx).