To add to the above, I recommend spending a night at the Mammoth Mountain Inn. It's 9000 feet up and has full hotel amenities and comfortable beds. It's tough getting a good night's sleep at altitude and without a comfortable bed, but if you have one of the two (comfortable bed in this case), then you can sleep fairly good.
I value getting good sleep just as much as acclimation, and a high altitude hotel will deliver both to you. Of course the day before the hike you want to hang out at the Whitney Portal campground, and it's only a 2 1/2 hour drive from Mammoth Mountain.
The nice thing is that the Inn is almost right on your way, maybe adding 5-10 miles to your trip (but breaking it up so you don't have drive 8 hours straight through to Lone Pine).
I wouldn't do anything more strenuous than a 5-6 mile warm up hike like going up to Lone Pine lake. My wife and I just hike from the campground to the Whitney Store and that's just to kill time more than anything else. The main thing is logging hours at altitude and the longer and higher the better.
