or "Asteroid of DOOM!!".
If any one of the largest 5 chunks had hit earth, we would not be having this conversation. If it didn't kill you outright, you likely will have eventually starved. I say this as a farmer with a garden and the ability to can food. I'd be lucky to have food on the table a few years later. Those of you in cities would have most likely starved. Our infrastructure for producing and delivering food is to fragile, and stretched to the limit. The medium and smaller fragments likely would have caused famines, but would not have been civilization ending.
The moon is to close. It would be peppered by huge numbers of meteors thrown up from the impact. Any colony there would be wiped out by them. Same goes for any space stations in earth orbit. That leaves the asteroid belt, moons around other planets, and other planets.
Mars has the raw materials needed for building a colony all in one place. The asteroids may not. Once Mars is self sufficient, it doesn't need space capabilities. Any colonization of the asteroid belt will forever need space ships. At the minimum to divert asteroids on collision courses with the habitats. That means fuel to propel them. Where will you get it? Shipping fuel around requires lots of fuel.
Yeah, any self sufficient colony is a multi generational project, but it must be started some time. We don't have a choice. That clip of Comet Shoemaker–Levy 9 hitting Jupiter in 1994 is a warning. BTW, in 2009 another slightly smaller sized impact hit Jupiter. That time nobody knew it was going to happen. We only got to see the pacific ocean sized scar left behind.
So what is worse...
Aiming low and hitting your target
or
Aiming high and possibly missing - but hitting a mark higher than where you currently are?
Always go for batting the ball out of the heliosphere.
So far NASA is the only one to have done it. Voyager 1 left the heliosphere back in 2012.