There's a lot of hype about colonizing and terraforming Mars. Wouldn't it be cheaper and more feasible to do the same thing on our Moon?

Great question! I actually did some work at NASA-JPL to see how Martian (Mars) missions could leverage off of Lunar (moon) missions because there are some synergies between the two. However, the differences between the moon and Mars are such that they are not exactly the same when it comes to colonizing and terraforming. First, let us take a look at the differences for terraforming:

Terraforming generally involves the process of converting a planet (or other body) to be more like Earth, in an attempt to make it more habitible for life originating on Earth (not just humans, since humans need other plants and animals to survive). For a place like Mars, the terrraforming process would involve raising the surface temperature to a more balmy climate. There are several ways to do this, none of which really need to be discusses or debated for the purposes of this article. Now, here is a really cool thing: Mars' atmosphere is mostly carbon dioxide. Now, it is at a really low pressure, but if you raise that pressure and maintain the concentration of carbon dioxide, the planet will begin to warm and you can start doing things with plants. Plants consume carbon dioxide and put off oxygen. This is basically the process that happened on Earth eons ago, with the plants converting the mostly CO2 atmosphere to something more akin to what we have today. So, if you raise the pressure of the CO2 atmosphere on Mars, it will raise the temperature (think "global warming"), and subsequently allow for the use of plants to create an oxygen-rich atmosphere for other life. Basically, you end up with a pretty simple terraforming plan.

Compare this to the moon, though. The moon has no atmosphere, carbon dioxide or otherwise. So, if you wanted to terraform the moon, you would need to build an atmosphere. This is not an easy process at all, pretty much impossible.

It will be much easier to terraform Mars, making it into a place where life from Earth can take up refuge and continue on.

Now, colonization. The moon, in this case, will be far easier to colonize. It is quite a bit closer than Mars, which has a number of positive factors for colonization. The closeness to Earth means that you can colonize the moon pretty quickly, without having to deal with long travel and signal delays. And, once the colony is up and running, there is only about second of latency for communication with Earth. Compare this to almost 45-seconds of latency at times for Mars. Just being closer means that communication between the Earth and its moon will be much simpler for a colony.

So, in some ways, the answer to your question is "yes." It would be simpler and cheaper and more feasible to colonize the moon. But, if you are looking for a more permanant solution, then terraforming Mars, which will work a lot better, is the way to go.