Southern/Central California

In Yosemite/Sequoia you will need to protect your food from bears. You can rent bear canisters from the park or you can stay on the more popular trails as most will have bear boxes. June may be a bit early for the high country as it will depend on the snow melt. If it's not accessibility due to snow pack then it may be due to the rivers running too high. BTW, if you do not have permits and they are completely book (which is more than likely the case, you will have to go on a first come first serve basis. I think they hold about 50% of the slots for first come first serve. The most to least popular are; Yosemite, Sequoia-Kings Canyon and you may consider Mineral King. The valley floor in Yosemite can be an absolute zoo in the summer.

Hamilton Lake – Sequoia NP
Grand Canyon of Tuolumne River – Yosemite

I second the idea of doing the Grand Canyon of the Tuolumne (Yosemite). With this year's snow pattern, June should be the perfect month to do this hike. (Unless you are talking June 1; that's an iffy date anywhere in the Sierra backcountry.) The elevation is 4000-9000 feet. June is too early for most hikes in the true high country, 10,000+ feet.

It's about 30 miles. Park your car at Tuolumne Meadows (Lembert Dome) and hitchhike to White Wolf. (You can spend the night at the Tuolumne Meadows or the White Wolf Backpacker's Campground for $5, if you have a permit to start the next day.) Ten miles on Day 1 will get you to Pate Valley; ten miles the next gets you to the first of the huge waterfalls. Third day to the car.

The White Wolf trailhead is not one of the really popular ones in Yosemite. That's good, as most likely it is still not reserved up for June. Snap it up.

– Elizabeth

Don't forget to take into account the altitude and climb if you go to the Sierras. I always find the altitude and dry air to take more out of me than I expect. Depends on your fitness level of course but as a transplant myself it took me a while to adapt my trip mileage down a bit to take this into account.

Thanks for the advice everyone. I think the plan is actually going to involve a journey to Mineral King and some combination of available trails in that area. Reservations for that area are completely wide open right now so it should be no problem.

I'm hoping that the snowpack will be fully melted and the ground hardened by June 18th or so but I'm not sure how the snowfall was this season in the Sierra, anybody have a conjectural estimate on this matter?