From St George, Utah to Fresno, the shortest way is via Las Vegas...Yahoo says that is an 8 hour drive. Unfortunately, there just isn't any camping close to Fresno. You could go to Sequoia NP on Monday and leave very early Tuesday to get to SMB.
From there, you could head to Yosemite via Highway 41 and hopefully find a campsite as Joanna suggests. After checking out the Valley, head up to Tuolumne Meadows and Tioga Pass to cross the Sierras. Go south on 395...there are lots and lots of places to boondock along this highway. If you every watched old westerns from the 50s, there is an interesting area just west of Lone Pine where many of these movies were shot. From 395, you can go through Death Valley and head toward the Grand Canyon. We haven't done it, but I've heard good reports about camping at Toroweap on the north rim of the canyon.
'01 Ford EB50p Quigley 4WD