You have a lot of options.
Free Yosemite Valley shuttles access nearly everything in the valley. Free Tuolumne Meadows shuttles offer similar, if less frequent, access to trailheads and other important points along Hwy. 120 in Yosemite’s high country. Their schedule is posted online:
http://www.nps.gov/yose/planyourvisit/tmbus.htm
Stay west of the park and YARTS will get you back and forth — from the Bug Hostel, Mariposa, or even Merced if you want to commute that far. It does six or eight roundtrips each day (weekday vs. weekend), and except for Merced costs $12.00 or less, round trip, depending on distance traveled.
The YARTS bus from the Valley to Tuolumne Meadows continues on to Lee Vinning and Mammoth Lakes (east of the park). It runs once a day each way: afternoons, east towards Tuolumne; mornings west towards the Valley. You can also take it to and from, say Porcupine Creek campground to do an overnighter to North Dome.
http://www.yarts.com/schedule.html
A backcountry permit (departing from any Tioga Road / Tuolumne trailhead), includes access to the Tuolumne Meadows backpacker’s campground: $5.00 a night, no reservations needed. You can stay there one night before and one night after your hike. You could take the afternoon YARTS bus to Tuolumne, spend the night at the Backpacker’s camp, and leave the next morning for Cathedral Lakes or Sunrise or Clouds Rest or anywhere else in that general area. You could then spend two or three days hiking back to the Valley. Or just make it an one-night trek, spend the next night at the backpacker’s camp again, and take the morning bus back to the Valley.
On past trips I’ve had a rental — parked it in the backpackers lot on arrival and retrieved it on the day I left. This year I’m taking public transport from the SFO airport. So I’ll know in mid-July how much of a problem going car-less really is.