I did it after undergrad. It takes very little planning. I honestly had never heard of the AT, saw a Backpacker magazine in the airport, and three weeks later was on the trail in Georgia. Read up on some gear, buy the good stuff not the shitty cheap stuff, and just go do it. You will want to start in the spring in Georgia, between the middle of March and the middle of April. If you're young and fit it will take you 4-5 months most likely. It can be done faster, and some do it slower, but you will most likely fall between 4 and 5 months.
It is an amazing experience. It does not define my life, as so many say it does. It is simply a part of a very active, diverse life. Plan on taking approximately $3K for your time on the trail for food, hostels/hotels, gear replacement, etc.
If you want to geek out on everything AT, go to whiteblaze.net. Just be prepared to wade through a plethora of old guys who want nothing more than to tell you exactly how to do everything, when to do everything, and why it works/worked for them.
A suggestion I have is to go thru-hike another, shorter trail beforehand. Consider the Long Trail in Vermont (shares 100 miles with the AT). There are dozens of other trails that fit the bill. The John Muir Trail is fantastic as well.
