There is NOT a lottery for JMT permits starting in Yosemite and hiking SOUTHBOUND. For southbound you get your permit one of two ways: 1) fax in the form in advance when the permit window opens and select your dates and alternates, number of people in your party,etc. With the glut of people doing this, you may or may not be successful in getting your #1 choice of dates or even a permit. The only thing you can do to increase your chance of success is to submit the form immediately when the permit request window opens and have afew alternate dates.
(Since you have three people, I would recommend submitting an application for an advance permit.)
2) The second way to get a permit is to walk-in. As already noted, there are a certain percentage of permits held for walk-ins, but you need to queue up early at about 0400 to get in line to compete for the walk-in permits. Also, cancellations or no-shows from reserved permit-holders are made available at about noon to walk-in customers. Walk-in permits are distributed as "first-come, first-served."
If you score a walk-in permit it will probably be administered to officially start the JMT the following day, with a same day permit to camp at Yosemite Valley, which is along the JMT route anyway, so in effect you are starting that day. Your JMT permit issued here covers you for the entire route, including of course Mt. Whitney and out Whitney Portal.
If you are doing the JMT NORTHBOUND, then as Bob and other posters have said, you need to compete in the Whitney permit lottery to start at Whitney Portal. But again, since you're going SO, you don't need to worry about this.
Have a great trip.