Alpinestyle:
Suggest that you think about climbing Whitney via the Mountaineers Route. It is somewhat more technical, less crowded and more interesting, in my view, than the main trail. In order to climb it, you do not need a permit for a one-day climb, but do need one if you plan to bivvy on the mountain. You can do a very Shasta-like climb by climbing to either Lower (~10,500') or Upper (~11,500') Boy Scout Lake to bivvy, then do the summit the next day from there. There are only a couple of sections where you may need an ice ax and crampons this time of year. There are also a couple of short and one long section of class 3 (some say class 4) climbing. If you do a search on this board for Mountaineers Route, you will find good route descriptions and pictures.