The best place I've found is Etienne on Division Street. They're not expensive, and they're really friendly in there. I broke my wrist a while back and couldn't dye my hair, and they dyed my hair using my own hair dye for me, and then they styled it really nicely too. Plus they're the only hair dressers I've been to where when I ask for the ends to be trimmed, they just trim the ends, not take an inch or so off!!
I've been to many hairdressers around Sheffield, and they are by far the best I've been to. They get my full recommendation.