Definitely check out the University of Kentucky. I am a freshman classics major at UK, and I turned down a full tuition scholarship at USC to take advantage of UK’s amazing classics program. What’s unique about it is that there are 500-level classes taught entirely in Latin (yes, the professors speak Latin fluently!) as well as informal conversation practice sessions which anyone, regardless of level, is welcome to attend. The Greek professors are trying to take their program in a similar direction. Just this year, a grad student started weekly conversational Greek sessions which I have been attending. The website for the classics program is http://mcl.as.uky.edu/classics, and the page for conversational Latin opportunities specifically is http://mcl.as.uky.edu/latin-institute. The classics professors here are also some of the kindest people I have ever met. They love Latin, and they love their students.
As for grad school admissions, the directors of the UK Latin Institute, Drs. Tunberg and Minkova, are fairly well-known in the Classics world, so I think a letter of recommendation from one of them would be looked upon well by graduate schools. Also, for highly motivated students with at least 15 hrs. of AP, IB, or Dual credits, UK has a 4-year BA and MA program in Classics that would also be a plus for Ph.D program admissions and would speed up the process as well.
Also, UK has some pretty great merit scholarships. Full tuition, room and board, $1000 stipend, $2000 study abroad grant, and iPad guaranteed for NMFs, and similar packages for competitive merit scholarship recipients.