Read the wilderness permit page on the Yosemite website,
www.nps.gov/yose
Follow directions.
There is a lottery for Half Dome permits. Not backpacking permits. 40% of the wilderness permits are held back for walk ins, which start the day BEFORE the date of the permit. For very popular trailheads you will have to show up early at the wilderness office the day before, and stand in line, for a walk in. If you are very lucky with a fax machine or getting through by phone, 168 days prior to your (hoped) starting date, you can reserve the permit of your dreams and be able to pick it up any time the day before OR the day of your hike, as long as you get there before 10 am -or- you call them the day before and tell them you will be there after 10 am (if you can't get there before). If you don't pick up a reserved permit it is handed out to the next walk in who asks for it.
So, there are three chances to get your trailhead. Reserve it - if not successful, try for walk in permit the day before. If not successful, show up at the wilderness office the next day and be there on the dot for when the abandoned reserved permits come available. IF you are 1-2 people, show up in the middle of the week, and willing to sit for a few hours to be in the first 6-10 people in line, you have a good chance of getting a walk in.
I spent about three weeks one summer walking in, getting a permit in Tuolumne, backpacking for a few days, returning, getting a walk in permit, resupplying, backpacking for a few days, getting a new permit, riding the YARTS somewhere else, hiking back to Tuolumne, backpacking for a few days.... I had a walk in for the Grand Canyon of the Tuolumne, arguably one of the more popular trips in the park. Figure into your plans that you can stay in a backpacker campsite the night BEFORE and for one night AFTER the dates on your permit, and if you arrive via public transportation you can stay more than one night - this is in the compendium tho some rangers do not appear to know about it, I had a friend get thrown out of the Valley BP camp even though this is the rule, may want a printout of that page.... Backpacker campground charges $5/person per night. I had a lovely time being a hobo and sneaking into showers at the High Sierra camps here and there.
On the wilderness page, there is also a list of trailheads organized into various areas of the park, and a full trailheads (all reservable permits are reserved) report. These will be helpful in the event you would like to reserve anything.
I recommend getting one of the many hiking/backpacking guidebooks at a local library, or visiting Amazon, to read about the various routes and places. Or some of the trip reports on here - or some of the blogs around the internet. Googling names of trails will get you a lot of information.
I know nothing about this so....Are there drawings (lottery) to backpack YNP? Are the permits difficult (next to impossible?) to get.