Danny,
Great question! It’s about 200 miles from LA to Lone Pine. From Lone Pine you can hitch a ride up to Whitney Portal, then hike up Whitney… but you’d need a permit. You might be able to get a walk-in permit. This is the shortest route to the southernmost part of the John Muir Trail.
Getting to Lone Pine you can either drive up to Lone Pine (actually you’d park at Whitney Portal). Parking there is free.
You could also take the Antelope Valley Metro line from Union Station to Lancaster, then pick up the Eastern Sierra Transit Authority bus to Lone Pine.
If you can’t get permits for Whitney, consider starting at Horseshoe Meadows, just south of Cottonwood Lakes. This will add another day, but it’s much easier to get permits for.
As for “the best week” do you mean time of the year or the section of the trail?