With all the snow over the past two years, you might have problems where ever you go, unless its in late Aug, Sept or later.
Think about going to Pacific Valley, off Hwy 4, and there are several nice hiking trails from there. There's a great trail that goes to Bull Run Pk. Its an easy drive, there are nice, free campsites, and its beautiful.
If you've been searching for the best source of information and stimulating discussion related to Spring/Summer/Fall backpacking, hiking and camping in the Sierra Nevada...look no further!