First Step: The Moon

Source of many of the quotes below: Why Don’t We Try To Colonize The Moon Instead Of Mars?
Even though its atmosphere doesn’t sustain pressurized oxygen like ours, which is imperative to live, it still protects the planet from the destructive shower of meteoroids, a phenomenon ubiquitous on the moon.
So when people SAY "atmosphere," laymen HEAR "We can BREATHE there," NOT "Rocks won't smack us in the head from space because they burn up in the atmosphere."

My understanding is that currently our best bet for colonizing the Moon is by establishing a colony in underground caves. Ergo, we MOSTLY won't care about rocks hitting the surface, with the possible exception that if we have skylights or something to access sunlight for growing food and other plants, we will want shield doors. Instead of monitoring weather, we will have a meteor report so we can plan to close the shield doors.

We can potentially establish a breathable atmosphere in "pockets" underground on the Moon using microbes. Mars would ALSO need to somehow solve the breathable atmosphere problem. Regardless of which rock we aim for first, this is a problem that MUST be solved. It's not necessarily easier on one than the other, just different.

The latest info I have indicates that BOTH the Moon and Mars likely have SOME water. The moon is small enough that the water it has MAY allow us to begin the terraforming process largely using resources on site -- potentially including developing an atmosphere because water contains oxygen.
Mars’ gravity is less than Earth but large enough to lure gases and form an atmosphere. Its gravity is one-third than gravity is on Earth, a slump in weight that will require its colonists some time to get accustomed to. Compare that time to the protracted time colonists of the moon would require getting accustomed to its gravity, which is just one-sixth of gravity on Earth.
So it's farther to travel to Mars and it STILL has substantially LESS gravity than Earth. Meanwhile, we can use the moon as a lab for testing weighted suits, workouts, and other solutions to the consequences of living in low G.

So far, we've dealt with astronauts in WEIGHTLESSNESS. Do we even yet KNOW what it does to the body to be in extended low grav instead of WEIGHTLESSNESS? I doubt it.
Furthermore, the cratered and craggy surface of the moon is thoroughly suffused by lunar dust. The abrasive powder clings to everything it acquaints. Known as regolith, the dust is fine enough to slip through narrow crevices and cause mechanical failures. These failures could cause permanent damage to expensive exploration machinery and cost us entire missions, but more importantly, a life.
Very funny. We have regolith on Earth. It's the basis for how rock begins to become soil and in some ecosystems is a critical part of the ecosystem.
In upland granitic terrain in California, thin soils overlie a thick zone of saprock. Although soil has a greater water- holding capacity, the saprock, because of its greater thickness, constitutes the greatest reservoir of plant-available water. For example, in a Jeffrey pine forest in the southern Sierra Nevada, the regolith consists of an upper 75-cm-thick layer of soil with a plant-available water capacity (PAWC) of 20% that overlies a 275-cm-thick layer of saprock with a PAWC of 12%. The result is that the soil retains 15 cm of water in its 75 cm thickness, whereas the saprock holds more than twice this amount (33 cm). Since this forest site loses at least 40 cm of water by evapotranspiration annually, mostly during the summer dry season, water stored within the soil cannot support the water demands of the forest (Rose et al., 2003). For example, in 1996, plant-available water in the soil was depleted by the end of June (Fig. 5), and the plants had to rely on water stored within the saprock for the remainder of the summer dry season (which extended to the end of October). (source)


Furthermore, desert sandstorms can be a big problem in some places on Earth. Dust getting into everything is NOT a problem we have never dealt with before.
The most promising dwelling place seems to be the lunar poles, which are not only believed to rest upon large slabs of ice, but are also subject to comparatively calmer temperatures. The low levels of sunlight falling on them for prolonged periods maintain temperatures of up to 32 degrees Fahrenheit. Rather than a blazing lunar noon, the poles experience more of a solemn sunset.
And here we have it: We already know of areas on the Moon with more friendly microclimates.
...scientists are still uncertain about the presence of water on the moon.
This just in: Scientists Find Water in Glass Beads From the Moon
This means the lunar surface could hold up to 300 billion tons of water, a new study estimates
Even though the benefits of terraforming the moon might outweigh its risks, scientists continue to dismiss it because, at the end of the day, it is a cold, harsh, barren and dangerous place to live. Considering the travel costs and convenience, the moon seems to be the ideal place to settle economically. In the long run, however, Mars emerges as the clear winner.
In reality, we need to colonize the Moon as a LAB for working out how to colonize Mars. The logistics of "cold calling" Mars simply don't make sense.

The cost of failures in figuring out HOW to do this will be dramatically worse if we try to colonize Mars first. Those costs will not just be financial. They will come in lives lost, opportunities lost, time lost and the psychological toll of continuing to try in the face of no immediate pay-off.

Research for how to colonize the Moon is much more likely to have immediate applications for improving life here on Earth and the costs of failure -- and there WILL be failures -- will be much less.