If you have limited time, the hike to Cathedral Lake gives you a sample of
what the Yosemite Backcountry offers.
There is also an upper lake that requires a bit more time.
The trailhead to the Cathedral Lakes is on Tioga Road near the western end of Tuolumne Meadows, about 14 km west of the Tioga Pass entrance or 61 km east of the Road 120 entrance at Crane Flat (past Tenaya Lake).
The distance from the (well-marked) trailhead on Tioga Rd to Cathedral Lake is advertised as about 5km, although i think it is more like 6km. The trail is the wide John Muir trail until the well-marked junction with the trail that heads down into the meadows and then to the lake itself in about 1km.
The loop to the Lower and Upper Cathedral Lakes is about 13km.
Trailhead Elevation: 2,590 m