I've done several backpacking trips along the eastern Sierra's and they are all spectacular. Whitney is popular because it is the tallest, but everything around it is nearly as tall.
Here is the list of trailheads in the Inyo National Forest:
http://www.fs.usda.gov/activity/inyo/recreation/hiking/?recid=20228&actid=51more good info here:
http://www.fs.usda.gov/Internet/FSE_DOCUMENTS/fsbdev3_003592.pdfMost of those trails can be viewed with contour maps on google maps. I suggest looking for trailheads that start at higher elevation, that makes it easier to get up high.
Hilton Lakes is a good beginner trek. If you get tuckered out early you can stop at Davis Lake. The more energy you have the higher you can go up the chain of Hilton Lakes.
If you want something closer to home, you can do sections of the PCT which goes right by LA. Last weekend I did some day hikes on the PCT up near Wrightwood, CA (Angeles NF). It is beautiful country though it seemed more dry than the parts of the Inyo NF that I have been to, so make sure you make adequate water plans (you can inquire at the ranger station about water sources along the trail).