The moon has roughly four times the surface area of the US and now most of it is prime timber/farmland. New colonization era ho. The Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space is going to last approx. 0 seconds with something that valuable on the line. I'd expect every nation that can launch to be working towards grabbing them some regolith as fast as possible and a lot of sabotage and maneuvering will ensue. Ultimately China, Russia, the EU, and the US will all put boots on the ground and claim them some land. India may be able to pull it off before all the land is claimed if they're willing to blow a massive chunk of budget on getting their fledgling program into high gear. Iran will claim they're going to launch and build a mockup of their planned lander. Canada has a decent program for people and know-how but doesn't have any actual rockets as they normally piggyback on European and US launches, if they can make a good case they might get Canadian colonists on somebody else's launch but I wouldn't bet on it at this point as the prospect of this much land is going to generate some greed.
Of course how many launches a nation can do is going to matter, a lot for establishing colonies. China looks like it has the most launches but their payloads are small and they're close to the bottom for actual crewed vehicles. That can change as pretty much everybody's going to be building max-crew capable vehicles as fast as possible to get boots on (claiming) the ground.
Apollo 11 and 15 look like the best sites to me, being close enough to readily build ocean ports which are going to be critical for initial food from fishing and easy transport, as well as letting the first colony readily spread out and claim more territory. As sea travel is simply faster and easier than land (and provides a steady supply of food from fishing) the initial colonized area will be the seashores of the main ocean, then colonists heading upriver and claiming land there with the sections in-between and viable only to land vehicles coming in third as forest gets cleared and roads are built.
One odd side effect is the Lunarians may actually develop into a different strain of humanity. Given the extreme expense of launching a person, most countries are going to be sending, well, astronauts, which is to say people who are both physically in the top percentile of health and geniuses as well. With a starting population of humanity's absolute best as the initial breeding pool and a significant time before any others show up, it's not impossible that Luna could wind up creating actual supermen. Of course it's also possible that what we think of as inferior genetics, weaker bodies and such, actually have unforeseen benefits and the lack of weak, stupid people actually hoses the genepool. Either way it would give us some interesting insights into human genetics... and also probably create racism issues that make everything we have now look like a circle singing Kumbaya.