The main problem with colonizing places is the displacement of the people already living there. You'll notice that space is notorious for not having people. It's one of the defining traits of space, really.

As to staying where we are, well. That comes with all sorts of issues. The first of which are big rocks. Then there's gamma ray bursts, and coronal mass ejections, and a host of other potentially life ending things that could hit our planet at any time.

We have all of our eggs in one basket. This is the height of stupidity when we could do something about it.

As to fixing our own planet? Why the fuck do you think we can't also work on that? There are billions of humans, we can surely multitask. Especially since actually living on the moon or Mars or whatnot will be a monumentally hard task in and of itself.

The first moon base will need to be 100% science to figure out some pretty important biology, like is it even possible to maintain a population at 1/6 Earth gravity.

That's a huge question that we don't actually have an answer for.