As others have noted, Inyo NF manages the wilderness permits from the Devils Postpile/Reds Meadow area:
If you have a specific date in mind, you should reserve a permit--the Devils Postpile area is popular and quotas can fill during peak season (summer/weekends). The website above has information on when permits can be reserved (generally, 6 months in advance). You can fax in a reservation request. You can also ask Inyo to have your permit held for you at a ranger station if you can't pick it up during their normal hours, as long as you're not going into Kings Canyon/Sequoia NP.
Heading north from Reds Meadow, you can take the "High Trail (PCT)", "JMT North", or the "River Trail"--all converge near Thousand Island Lake. Heading south, you'll want the "JMT/PCT South".
Note that if you plan on exiting via Whitney Portal, review the "Trail Crest (exit only)" information on the website--there's a separate quota to exit via the Main Whitney Trail. The challenge will of course be knowing which date you'll be exiting at Trail Crest, but I wouldn't expect any problems with the rangers if you make a good-faith estimate and are a day or two off.