Colonizing moons is done in a race's "local actions" menu in their political screen for pretty much every race. I'm pretty sure it's not exactly called that, but you'll likely see it in there anyway. There's an option in there to do it.
As for properties, if you mean buildings, that's in the "friendly actions" screen, it's one of the "dispatch" options. Meaning that it takes time to help them build whichever building it is.
Note that in order to be able to construct a building though, the race must have the tech that allows them to do so, whichever tech it is. There can be multiple buildings of the same type on a planet. Look at the "planetary sidebar" or the planet status screen, near the bottom of that listing of stuff there's a list of which buildings they already have in place, and how many of each.
If you mean outposts (scientific, construction, military), you have to have a race build one... either on their own, or with your help... and then once it's completed and showing up on the solar map screen near their planet, you must go and capture it for yourself. Needless to say, this causes a chunk of negative influence, but it's not THAT bad, so if you're in good straits with the race in question and have built up influence with them, it wont cause too much of a problem.
Hope that helps. If there's more specific questions about it, do feel free to ask!
