If you have limited time, the hike to Tuolumne Falls gives you a sample of
what the Yosemite backcountry offers.
The trailhead to the Tuolumne Falls is on Tioga Road 500 meters east of the visitor center, 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 Glen Aulin camp (the White Cascades) is less than 8kms. The trail follows the canyon of the Tuolumne River, so it loses elevation (trailhead Elevation: 2,590 m). One can return via McGee Lake to Tenaya Lake (Murphy Creek trailhead). This is about 11kms. From Tenaya Lake a trail coasts Tioga Rd to Tuolumne Meadows (about 14 kms). Or just arrange for a pickup at Tenaya Lake.
Note that you are heading downhill on your way to the waterfalls, and uphill on the way back.
If you have a whole day and can shuttle two cars, consider hiking from Tuolumne Meadows (elevation 2627 meters) to White Wolf (2365m), 50 kms on the trail that runs along the Grand Canyon of the Tuolumne river.