If you're asking how to get there without driving, then I believe the only choice you have is to stay in Visalia. There's an Amtrak station in Visalia - and a shuttle.
If, however, you're simply asking how to drive there, it would be a bit faster to fly into Burbank, if possible, rent a car and drive. Head north out of Los Angeles on the 5, taking the 99 through Bakersfield. You get off on the 198 (Signed to Sequoia National Park), go through Three Rivers (maybe you are planning to stay there? - that works) and then up the mountain!
If you go, do take the road to Crescent Meadow as well, where you will see big stands of trees to rival General Sherman. And even before that, you'll have seen trees that are simply amazing.
If you flew to LAX, spent one night recovering, then picked up your car, it would take you a reasonable 5 hours to get to the park the next day. You should have plenty of time. Look into staying in the park (Wuksachi), too.