You'll need a wilderness permit if you want to stay in the backpackers' camp in YV. See
https://www.nps.gov/yose/planyourvisit/wpres.htm. You can either reserve the permit in advance (check www.nps.gov/yose for availability) or get one on the spot (on the day you leave or from 11 AM the day before). Even if you reserve in advance, you'll need to pick up the permit in person. If don't reserve in advance and plan to get one the day you start your trip, you can expect to spend some time waiting in line to get the permit, so take that into account. If you've got the permit, you'll be allowed to leave your car in certain parking lots in YV. If not, and you're not staying in one of the campgrounds or at one of the hotels, I don't think there's anywhere to park overnight in YV, but there are others who can answer that question.
If you're planning to stay at one of the campgrounds (other than the backpackers' campground) or at a hotel, you'll need reservations; those generally sell out far in advance, but check with yosemithospitality.com and recreation.gov for availability. There's also Camp 4 in YV, which is a "walk-in" campground, i.e. first-come-first-served, but I don't know from what time you can try to get a spot there or how quickly it fills up.
Keep in mind that you'll have a (much) better chance for campground reservations and wilderness permits if you go mid-week. And there are some places outside the park on highway 140 and highway 49 where you can stay.
Permits are issued for leaving on a specific trailhead on a specific day and returning from a particular trailhead on a specific day. Subject to certain restrictions (see the website), what you do in between is up to you.
As for camping in the backcountry, on the same web page where you get info about wilderness permits there's also a map showing where you can camp (https://www.nps.gov/yose/planyourvisit/upload/wildernesstrailheads.pdf). Most of the backcountry is wide open to camping, but there are some restrictions (e.g. no camping between Yosemite Valley and Little Yosemite Valley, no camping in Little Yosemite Valley itself between the LYV campground and Merced Lake, no camping along Panorama Cliff).
Consider arriving early in the morning, hiking up to the upper falls, coming back down while the permit office is still open and getting a permit for the next day. Then you can camp at the backpackers' campground and start your backpacking trip the next day. Given what interests you, you could start from Glacier Point and hike from there to Nevada Falls, camp at the LYV campground or partly retrace your steps and then branch off into the Illiloutte creek basin (toward Mt. Starr King) where camping is permitted, then the next day go back down to the footbridge and back up to GP, and do the short hike to Sentinel on your way out. (As recent discussions here have noted, trying to cross Illilouette Creek not at the footbridge will probably be dangerous this May.) Or if you get a HD permit, continue past Nevada Falls and stay at LYV or closer to HD, start early on HD, come back down the John Muir trail to YV and then take the tourist bus back to your car at GP (but don't know the time of the last bus).
Lots of possibilities, poke around this site and you'll find lots of info. The most important thing (as JKW has said many times, and it can't be said enough) is to be flexible and have several alternative plans in mind. If you're a strong hiker and score a HD permit, you can do that as a day hike out of YV, so maybe a backpacking trip isn't necessary. If you don't get a HD permit and are a strong hiker, you could do upper YF and Nevada Falls in the same day, then do Sentinel and Glacier Point, and some other stuff (Taft, Dewey) the next day.
Edited 1 time(s). Last edit at 04/13/2017 05:48AM by Not quite The Geezer, but getting there.