The costs to simply lift a signficant portion of Earth's population into space, even if they did nothing other than die immediately*, would be staggering. And that's not even getting them into orbit, let alone to the Moon.
A few people have mentioned cost, but I don't think that's necessarily the biggest concern. More important is simply the number of people you would need to move. Let's assume that we've solved all the problems associated with building a lunar base and are ready to start shipping people out there. How many people do we need to move? Well, the current population growth rate is around
78 million people per year (134 births, 56 deaths). Deaths are expected to increase to 80 or so by 2040, so lets conservatively say the population is increasing by 50 million per year once we're ready to start shipping. That's 137,000 per day.
So, how many have we actually managed so far? According to
Wiki, 529 people have travelled into space according to the USA definition. That's just over 10 per year, or 500,000 times too low. Of course, the vast majority of those never actually got anywhere near the Moon.
And how many could we actually mange? Let's use the shuttle as a baseline, although it was never actually capable of getting anywhere near the Moon (the highest was around 35,000 km for geostationary transfer, over 10 times too low). The shuttle can carry a maximum of about 25,000 kg, so call it around 250 people (it's around 1,000 m
3, so should be big enough to fit them in, if rather cramped for a several day journey). That means something like 550 shuttle flights every single day. So far there have been a grand total of 135 shuttle flights in just under 30 years, split between 5 shuttles.
In other words, we would need, at an absolute minimum ignoring any need for servicing between flights, 88 times as many shuttles (4 day flight each way, so you need 8 times more than the 550) making 45,000 times as many flights. To put that in perspective, there have been a grand total of
6854 spacecraft ever launched. Just to keep the Earth's population constant, you'd need to launch more spacecraft every two weeks than humanity has managed in its entire history. And that's with incredibly unrealistic assumptions about what a shuttle can actually do, along with rather optimistic assumptions about how many people you need to move in the first place.
So yeah. Shipping people off the Earth != the answer.
I think you are underestimating how far away the moon is. Depending on where it is in its orbit, the moon is between 362,000 and 405,000 kilometres away from the Earth. To put that in perspective, that's about 30 times the diameter of the Earth.
To put that in even more perspective, Jupiter has a radius of just under 35,000 km. In other words, extending the Earth's atmosphere out to the Moon would require turning it into a gas giant over 10 times the radius (and therefore 1000 times the volume) of the largest planet in the solar system. To put it in even further perspective, the Sun has a radius of a bit under 700,000 km. Extending the Earth's atmosphere out to the Moon would require turning it into a star half the size (1/8 the volume and presumably mass) of the Sun. The smallest known star has a mass around 1/100 that of the Sun.
So forget all the stuff about drag and pressure. What's being proposed here is having the most of the population of Earth live inside a star, with the remainder merely living at its surface.
Edit: One other thing worth bearing in mind. The Moon's surface area is a bit under 40 million km
2. Antarctica's surface area is around 14 million km
2. The Sahara's area is about 10 million km
2. Greenland is over 2 million. In fact, the Moon's entire surface area is only slightly larger than that of the Earth's 10 largest deserts combined. So if land is the problem, we already have plenty right here with virtually no-one living in it, and that is far easier to get to and far, far easier to make habitable.