If I understand it correctly, this will be your first mountain backpacking trip. In that case, let me make a suggestion for making this experience enjoyable.
In my mind, there are two very different ways to do this. One is to choose a popular high altitude route, on a heavily traveled trail to a heavily visited destination. Glen Aulin is a good example of this sort of thing. My first backpacking trip in Yosemite went in a bit farther on the first day, towards Waterwheel Falls. It was Fourth of July weekend in 1983, and we had been assured that summer in Yosemite meant that all we would need was warm (water-shedding) clothes, a sleeping bag, and a tarp. The first night it snowed. We nearly froze. But it was a memorable trip, and the falls were terrific. We were experienced backpackers from the East, so we had no trouble with the heavy packs or the cold-waeather camping.
The other way to begin mountain backpacking is with something simpler. If you are not very experienced, and aren't completely sure about wilderness cooking and sanitary arrangements and heavy packs and bear canisters and tents and squalor and rain squalls and all that, let me suggest that you make your first trip one where it is possible to disengage gracefully. To that end, I would suggest a short hike in to a base camp from which you can do interesting day excursions.
All Yosemite trailheads have a minimum distance that you must hike in before you can camp. See this map:
trailhead map link The more popular the route, the farther that you have to go. For example, from Tuolumne you would have to go to Glen Aulin or beyond, or well up one of the side trails. After driving to that altitude from flatland, without a chance to acclimate, that becomes a real grunt.
From Happy Isles or Glacier Point, the story is the same. But there are less popular trailheads where the required trek is much less. And some are at lower altitude. Here is one that I have used:
On the map, #1 is the Porcupine Flat Trailhead. That was our entry point. You only have to go past Porcupine Creek before you are allowed to camp. And it's downhill.
Here's the story. It's Fourth of July again, but 35 years or so after our Glen Aulin trip. Our college-age daughters and their husbands-to-be want some backpacking, but they want to bring a pair of female college friends who are interesting in trying out this sort of thing. They don't seem very outdoorsy, so I set up this trip. On the map, #2 shows the camp site we specified for the wilderness permit. But the mosquito situation in July can be bad near the creeks, so the scouts in our group found a possibly better camp site near #3. It's in an area of mixed forest and bare granite. It has a thigh-high shelf of rock for cooking, which the ladies liked. Plenty of logs for sitting, and running water not far away. Perfect.
As an experiment in backpacking it was also perfect. In the morning, the two guests packed up and left. Their packs had been too heavy, the ground had been too hard, there had been bugs and mosquitoes, and there were important TV programs that they were missing. Actually, I'm sure that the real problem was the little pit latrine that we had dug. The idea of squatting and pooping in the woods was just too dreadful. Undignified. They left in search of a motel with indoor plumbing.
After which the kids and my wife and I had several perfect days of side trips. It's an easy hike to Mt. Watkins. If you go there, be sure to visit the lookouts at #4 and #5, with their spectacular views of Clouds Rest, Half Dome, Pywiack Cascade, and Tenaya Canyon. Indian Rock and North Dome are nearby, or you can hike over for the view at Yosemite Point. On the way back from North Dome, you can visit Basket Dome and then walk home along the ridge to its north (see link:
basket dome). Airplane gully is also nearby (#6 link:
airplane), and the trail from Watkins saddle to Olmstead Point is an extremely scenic alternative.
Best of luck on your trip. If the short one works well, you can be sure of enjoying a more strenuous one later....