There are two (or more) schools of thought on city spacing. The experts mostly suggest going CxxC (that's City-space-space-City) and others suggest OCP (Optimal City Placement) which is spread out a bit more (CxxxC or even CxxxxC) to allow for full growth and the use of 21 worker tiles (the maximum for a city).
CxxC has several advantages. Cities are 3 movement points apart, so if a city is under attack reinforcements can get there in one turn (assuming roads are in place). There is no burning need for culture improvements as there are no gaps in your cultural boundaries (except maybe on border cities to reduce the threat of cultural flips). And most likely the game will be nearly over before you can research Sanitation which is needed to build hospitals in your cities to get above the size 12 limit. CxxC means almost all tiles in your empire are being worked by citizens, and thus no wasted tiles.
OCP would be for larger maps, where you expect your empire to be massive, the game to be longer, and you need massive production output. That's about the only benefit I see for OCP... I'll let others chime in with the benefits. A few of the problems are above size 12 cities start getting happiness issues (depending on your game difficulty) and early in the game you have tiles gaps that invite the AI to send in settlers. Also, for most of the game (until your culture border gaps close) there is a lot of your territory that is unproductive. And global warming is caused by population and city improvements, so OCP incurs more global warming than smaller cities with less improvements.
I personally play a bit of both. CxxC on the vertical and horizonal axis, and CxxxC when going diagonally or when terrain calls for it (ie coastline, volcanoes, or adapting to resources on the map). Sometimes I'll go CxC with an AI city if they are on my border and they are parked by a resource, hoping for a culture flip (I play religious tribes, so build lots of culture). In the very least they are one quick tile away for when I decide it's time to attack.