If the moon is colonized, it will be for mining purposes. Not mining of anything on the moon, but rather as a drop-point for automated asteroid mining, which presents an extremely rich amount of resources (both metal and carboniferous/water), and a construction base to actually build and launch the automated miners. At that point, the problem without a space-elevator (which I predict will be started but never finished) would be getting those resources to earth cheaply.
Then and only then will the moon really begin to be colonized, as the massive inflow of resources makes the quality of living there approachable to Earth's.
Attempting to colonize another planet(oid) for the purpose of relieving population problems is pointless, as the birth rate will almost certainly exceed the passenger capacity for the forseeable future, but at the same time if there
is overcrowding going on in developed countries with disposable income, there will be people that want to get away from it and go somewhere exotic. That would provide a small boost to the existing miner family population, further increasing the growth of any colony, but it would likely be quite small indeed. Colonies, if they occur, are going to be very poor at first, and very rich and exclusive towards the end.
These aren't all my predictions. Arthur C. Clarke I believe is the one responsible for turning me on to space mining
It makes a lot of sense when you consider that the fuel for a mining vehicle (which really could be any kind of mass at all) can be harvested at the source. And Red/Green/Blue Mars by Kim Stanley Robinson are the first books I've ever seen acknowledge the fact that it would take thousands of grueling and wasteful years to significantly decrease the population of Earth using colonization alone, if it's possible at all.