The traverse can be done in either direction from either trailhead, depending on how you prefer to do your climb. The Mesa Trail section (see Routes) is rather flat, so most prefer to get the hard vertical effort over with and finish the hike with the easier, flatter sections. If you're an animal, you may run the flat sections of the Mesa Trail first, and continue on to top both summits.
From Chautauqua Park, hike west for 0.6 miles along the flat trail that parallels Baseline Road. This will lead you to the Gregory Canyon Trailhead. From here, you have two choices for your ascent of Green Mountain. The Saddle Rock route is shorter and steeper, while Gregory Canyon is longer, easier, and somewhat less scenic (in the author's opinion).
For the Saddle Rock route, follow Saddle Rock Trail 1.4 miles to Greenman Trail, and follow the signs to the summit. For the Gregory Canyon route, ascend Gregory Canyon Trail for 1.4 miles to the intersection with the Ranger Trail. Follow the Ranger Trail and signs to the summit. Congratulations, you are now on top of Green Mountain, 8,144'. To the south looms the much larger Bear Peak.
Descend Green Mountain's West Ridge Trail 0.2 miles and turn left (south) and follow Green Bear Trail as it descends into Bear Canyon. After half a mile you'll reach the turnoff for Bear Peak's West Ridge. If you're feeling tired or bad weather is coming in, this is a good time to bail out down Bear Canyon Trail.
If you're still up for it, turn right, and follow the Bear Peak West Ridge Trail for a mile to the summit. The trail steepens as you reach the summit, and some moderate scrambling is required to reach the true summit. Descend the steep, rocky, and loose upper section of Fern Canyon Trail as it drops down Bear Peak's Northeast Ridge to a saddle at 7,500'. The trail leaves the ridge and drops into Fern Canyon. Continue down Fern Canyon Trail until you reach a wide gravel road. This is the Mesa Trail.
Follow the Mesa Trail north (left). Be careful to check signs at each trail intersection. After following the Mesa Trail for 2.4 miles as it winds across the feet of the mountains you'll find yourself back at Chautauqua Park.
According do the map, the route is 8.7 miles, and my GPS unit puts it at and even 10 miles. This route took me, an avid hiker, six hours to complete at a leisurely pace, with long stops on both summits. According to my GPS unit, I spent 4.5 hours moving, and 1.5 resting/etc. Plan for this hike to take the better part of half a day.
You could bring nothing but the running shoes on your feet, and be back at your car in about three hours. However, I would recommend a daypack with 2 Litres of water, some food, and a light wind/rain jacket.