Frank is right. Sequoia is not far from here. If you need info on the western Sierra's shoot me a PM. The area is outside your 1-2 hour drive limit, but all of the spots Frank mentioned are good. Here are some other contacts: Kern River, Kernville Fly Shop, Guy Jeans guide; Rivers north of there, Tule, Kaweah, Kings, San Joaquin, Merced, Jimmy Morales guide (he has a fly shop just outside the Yosemite entrance in Oakhurst, Ca.). Our fly fishing club has members from LAX even though quite a drive. You can check out the newsletters at Kaweah Fly Fishers
for outings. Most from the south LAX; Ventura head up the eastern side of the Sierras to the Owens River, or East Walker River. It's 5-6 hours for you.
Welcome to the Forum. Best to you in your search for some waters. Please keep us posted what you find.