Chris, I have been there several times hanging out with teams from our association who got "the call" As far as airports, I don't know I usually drive down, but I'm assuming Nashville as it's only about a 1 1/2 hour drive from there. As like any other comp set up is going on Friday, there is parking in and around Lynchburg. You will not be able to take a car into the comp area unless you have credentials. Things to do: Jack Daniels Distillery tour about a 2 1/2 hour tour; lunch at Miss Mary Bobo's Boarding House (a must!!! reservations required) visit the Lynchburg Hardware and General Store for all things Jack and souvenirs of the competition. When I have gone, we stayed in either Tullahoma or Manchester, Tullahoma is about a 20 minute drive to Lynchburg and Manchester is about 30-35. I highly recommend the Hampton Inn in Tullahoma, a little pricey but very nice hotel and comfy. In Manchester, we stayed at the Sleep Inn & Suites, very nice place. On Saturday there are a lot of people at the comp walking around...you must get there early is you hope to find parking in town. If not, there are a couple places outside of town that run shuttles into the comp site.
If I was you, I would arrive on Thursday, get your rental car, check into your hotel and then take a ride to Lynchburg and get the lay of the land. Friday...take take the tour in the morning or later in the afternoon and definitely hit Miss Mary's for lunch. Then spend the rest of the day walking through the comp area talking to teams, souvenir shopping, and take in the parade of teams later in the afternoon, etc. Saturday, get there early, watch the teams during turn ins, many teams offer samples to the crowd of their product. If you stay for the awards ceremony on Saturday, it begins at 4pm and it is long, probably about 2 hours or a little more. Once that is finished, go back to your hotel, pack and take a flight out on Sunday.
If you have any other questions, just let me know.