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.