Do your Private at a small or affordable part 61 school and then move on to a part 141 school after that to utilize your military benefits.
I will second the notion of staying away from RAA or DCA and looking into FSI. Wow, acronymn city...
Schedule a tour with them and ask about the veteran stuff. From what I hear from my former roomate at FSI (he was in the Marines) they go out of their way to accomidate former military guys.
Good luck!