Most hikers who spend a night climb the peak their second day, and hike out. But quite a few will camp a second night. If you want to take it leisurely, spend both nights at Outpost Camp. Just get up early and hike to the summit on your second day. If you are strong, you could camp one night at Trail Camp, and then hike all the way out after your summit on the second day.
99% of the Whitney hikers always exit the same place they start: Whitney Portal.
As for campgrounds, there are really only two: Outpost Camp and Trail Camp. Outpost is at 10k elevation, easier on the body for acclimatization. Trail Camp is at 12.5k, Makes getting to the summit easier.
