Found a great new site called Universe Today, which has an outstanding article on the challenges and opportunities provided by Lunar colonization. Just a sample of the reasons to go:
"the Moon is an ideal "staging post" for us to accumulate materials and manpower outside of the Earth's deep gravitational well. From the Moon we can send missions into deep space and ferry colonists to Mars. Tourists may also be interested in a short visit. Mining companies will no doubt want to set up camp there. The pursuit of science is also a major draw."
The biggest benefit in my view is that it would give us the practical experience to determine how to conduct long term exploration of other parts of the solar system, like Mars and the outer gas giant moons wher some awfully interesting things might be sitting. So what are the challenges? One would be to answer how materials hold up to long term exposure to the vacuum of space, extreme temperature variations, micrometeorite impacts, cosmic rays, solar wind particles, and other space hazzards.
How would we accomplish the establishment of a permanent outpost? First of all, we'd eventually have to use local materials like lunar regolith, likely for both the habitat itself as well as shielding from dangerous radiation and cosmic rays. The challenges of mining, drilling and excavating in a vacuum would also be considerable. There are several concepts that might provide the answer on how to get started, including enclosing a small impact crater with a dome, designing an inflatable or erectable structure, utilizing ancient lunar lava tubes or even mining out an underground habitat.
Once a base is established, then the work on a more permanent dwelling could be conducted and the settlement expanded over time. One thing I think would be a good way to start is to use a combination of these ideas - bring an inflatable structure but install it along the side of an impact crater and cover it with regolith, and use the space created to tunnel further under the surface.