[Roboshant]

FTC or VEX would be a great way to start for about 1-2k, although you can always subsidize that through fundraising and sponsors. Another option is Shell Eco-Marathon. Although it is not robotics, Shell Eco-Marathon provides for an engineering challenge of its own kind, and could be used as stepping stone to an eventual FRC team. I am the president of the SEM team at my school, and we actually managed to run our whole season on $400. SEM can be a very affordable program, as shell pays for transportation, lodging, food, and subsidize airplane tickets if needed. There are some really good SEM teams in NC, such as James B. Dudley High School (who seem to be very near to you), who would be willing to help you guys out. PM me for more details.

Edit: Believe it or not, but if you give yourself enough time a car can be built with only one meeting a week. We built our whole car in 7 days (our previous car was destroyed a week before the start of the competition so we had to build a new one).