There are many routes on Whitney from Class 1 to very technical vertical wall climbs.
You can choose from a "standard route" or create a longer trip starting from as far
away as Yosemite doing the John Muir Trail and making it a 200+ mile trip.
One option is to start from Kearsarge Pass which makes for a nice approach.
The most popular routes are the "Main Trail" and "Mountaineers Route"
For the Main Trail you can climb it all year, just be aware that the road getting to the
trailhead may pose a challenge if new snow falls making road dangerous or impassable
or is gated down lower, making the normal 22 mile / +6600ft roundtrip more like 30+ miles
depending on how far you can get up. This route is Class 1 all the way. Winter up there
can be brutally cold, well below zero with fierce winds or you cold have a nice 50*F day.
In the spring/early summer, you can do a good snow climb as an alternative route off
the Main Trail and go up the Couloir/chute to the west of the swichbacks from Trail Camp.
- makes for a a nice alternative to the switchbacks. I'd advise against glissading here
though since this slope is prone to ice up and the runout is filled with large rocks.
Mountaineers Route best done in either Spring snow or dry in the summer.
Doing that route in new loose snow and/or ice is not fun.
The MR is Class 3, some say Class 4 depending on your definition and travels
over a less defined trail from where you leave the Main Trail and while having
roughly the same gain, is a few miles shorter.
To clarify the permit situation - check with the Ranger Station at the Eastern Sierra
Visitor Center - I believe you need a permit for either
route. They used to separate the two permits but now think they combine them so the
same permit is used for either route ("Mountaineers Route" or "Main Trail").www.recreation.gov
is where the main resource is now for obtaining a permit for the area.
Check out the Whitneyportalstore.com or Whitneyzone.com sites for best current info.