The usual approach is via the Young Lakes trailhead, located at the eastern end of Tuolumne Meadows. Follow the signed trail for Lembert Dome around the backside of the dome, for no more than a mile or two at the most, to reach the granite slabs of the east slope.
An alternative option, but with limited parking (space for probably two cars at most), is to take the Dog Lake trail. This begins half a mile east of the Young Lakes trailhead along Tioga Road, and offers a slightly shorter approach to the east slopes of the dome. This trail is not shown on all maps of the park.
The official Lembert Dome trail ends at the granite slopes of the east end of the dome, but the final destination is obvious; the easiest option is to head slightly south of the main summit before working your way up the granite slopes to reach the top.
Expect company--even on a late October weekend, there were three or four other parties up there when I went. But for an hour or two's worth of time and very moderate effort, the views are superb.
A worthwhile stop on the way back is to head over to Dog Dome, located just northeast of Lembert Dome, for a different view of Lembert Dome and Tuolumne Meadows.
None; sandals would suffice.