When talking to the volunteers at the national forest information center in Rico, the most highly-recommended hike was the Cross Mountain Trail to the base of Lizard Head Peak. This hike has great views, a reasonable climb, and lots of wildflowers in July. Much of the hike is through the Lizard Head Wilderness, and bikes are not allowed.
The trailhead for this hike is just a few miles south of Lizard Head Pass, and there is an ample parking area with vault toilets. The trail meanders through meadows and conifers with increasingly beautiful views of Mount Wilson, Cross Mountain, and the uniquely-shaped spire of Lizard Head Peak up ahead. The trail ends at the junction with Lizard Head Trail, but the best views are from the top of the saddle about a half mile from the end. The hike back down presents another vista to the east of Sheep Mountain and San Miguel Peak.