Yosemite is a big topic. The massive numbers of people that flock there make it troublesome to do spontaneous backcountry hikes in the region. In order to see the popular areas, you usually have to plan far in advance; in order to see the less popular backcountry (although much of it is just as glorious), you have to know where you want to go, which means you have to do your homework.

If I were you, I'd drop my REI or other favorite backpacking store and pick up a few wide-area topo maps, and a few guidebooks to the region. You might consider other high Sierra areas besides Yosemite, such as Ansel Adams Wilderness area, or even farther south to Kings Canyon / Sequoia National Park. Then paw through the trail descriptions and examine the maps and come up with specific ideas, ask about those. Or goggle 'em -- there are dozens of sites dedicated to Yosemite et al, and thousands of web posts besides those.

A few other points:
* If you expect to go on a weekend, your options will be even more severely curtailed.
* The Yosemite NPS site has some crucial info: here's one I try to keep bookmarked: yosemite.org/rptFullTrailheadDates.htm -- that is the report of trailhead reservations that *cannot* be made; in other words, they're full. For example, if you find "Happy Isles->Little Yosemite Valley" you'll see that it's been full long since: that's the primary backcountry allocation from Yosemite Valley (Happy Isles is the trailhead at the SE end of Yosemite Valley) up towards Half Dome, Vernal Falls, etc. So once you know what trail you are interested in, you look it up via the trailhead that gets you in.
* They just opened Tioga Pass Road today, which should indicated to you how much snow is still in the high country. The volume of water is still high, so June is a great time to see waterfalls and streams. But if you want to go swimming in a tarn up in the granite, it'll still be mostly ice water until much later in the year (and mosquitoes will be more abundant as well).
* Some people ignore the rules (even skipping the reservation system and just heading in), but the rangers can make you pretty miserable if they catch you. I know some chaps that went into the Tenaya Lakes area without a permit, got caught, and were escorted out by the ranger to a fine at the end of the trail.
* Be aware that bear cannisters are mandatory in much of Yosemite, which adds some weight and hassle. Luckily, in Yosemite you can rent your cannister at one place and return it at another; at Kings Canyon/Sequoia you have to return them to the same place you rented them.

For example, I'm looking at the Ten Lakes trail, or maybe the Waterwheel Falls -- but I don't know those areas yet, so I'll be asking in another post in a minute about likely campsites. Those are my top choices after several evenings of pouring over my topo maps and guide books (and a little Google Earth).

Enjoy!

I have backpacked in Yellowstone many years ago, so I will not be able to give you trail/directions. I remember the route we took led us to a place called 'Mr bubbly' it was amazing. It is a point of convergence of a cold river, a hot river, and a geyser. It is a swimming hole and you get bubbles from the geyser. Several things i remember, they were very strict on where we camped the night, unusual since we were used to Yosemite and it is not like that (so regimented). Something else, Grizzlies. Be very careful (we were camping in an obvious Grizzly territory). Just make sure you take your clothes off that you used for cooking when getting in your tent for the night.
Anyway, not trying to discourage you, it is an awesome place and i hope to return some day. Have a great time!
(Ill post again if I remember something else)

I went to Yellowstone backpacking once about 4 or 5 years ago.


Just stop by the rangers office when you get your backcountry permit and ask them for recommendations on which trail to take. Tell them what level of difficulty that you want (distance, elevation gain, etc.).