I usually space 4-5 tiles away. It's really dependent upon the terrain. Valuable terrain? More spacing. Less valuable? Less terrain then shift to a specialist city once the available terrain is fleshed out.
I once loved to spread my cities out, particularly in Civ 4, but with 5, I've shifted my priorities. I focus on growth and production. That brings me money and science. I don't worry about culture or religion too much, as it seems to come in bunches anyway.
I try to give my cities a 2 tile radius. Three when in fertile land. Beyond that, I shoot for maximum food surplus then use specialists according to what the city is best at or what I need. Growth and production is key. More growth = more science to out-tech the enemy, and production to create the buildings required to give you that edge, as well as an army. Opposing civs lick your boots when you have a strong army.
Keeping your cities closer together allows for easy defense, less road maintenance, and less travel time. It is not necessary to have huge cites (24+ pop). It is far more beneficial to have more cities that are well defended. My goal is to be able to shift my garrisoned troops across my entire front as quickly as possible, reducing the need for extra small armies, which in turn allows for more $$$.