Finding a new barber that meets your expectations is just about the hardest things about moving to a new city, especially with a short cut presence... you're looking for a place you'll visit several times in a year. I'm also partial to the straight razor/straight edge cuts, something experienced old barbers do or expensive young barbers.
The College Barbershop on State has a mix of young and old and has been recognized for their skill (by Men's magazine I believe). I've gotten my cut by two different barbers so far, both young. I left satisfied with my vague haircut description and the result that they left me with.
Both did the straight edge cut, though, you need to be specific if you want more than just the back of the neck. The price was very fair too, coming in at just $15. The only thing is - its cash only, no cards, no checks... make sure you come prepared.
I'm hoping to see what the older guys can do on one of my next visits. Needless to say, I found my Madison barbershop!