Earth has a nasty habit of recycling its surface such that old stuff (like dinosaur bones and meteor craters and ancient alien cities) get swallowed up by oceans and volcanoes or eroded into dust. The moon, however, hasn't been geologically active for a very long time, and if aliens ever stopped by our solar system, the moon might be the place to check for artifacts.
If aliens ever did visit our solar system, odds are it was a while ago. Humans haven't been alive and kicking for all that long (geologically speaking), so while we have no idea what the chances of alien tourists actually are, we can say with relative certainty that it's much more likely that they were here sometime within the last billion years than it is that they were here last week.
While any billion year-old alien truck stops on Earth would have crumbled into dust a looong time ago, nothing has been going on on the moon for at least that long, and a recently published journal article suggests that this might make the moon the ideal location on which to spot hard evidence for aliens. As far as we know, the twelve people who have actually been to the moon didn't run across any monoliths or anything like that, but they were only able to see a tiny little bit of the lunar surface, and there's a lot left to explore. Currently, we've got a camera-equipped satellite (the Lunar Reconnaissance Orbiter) sending back lots and lots of high-res images of the moon, which you (anyone, in fact) can scan for telltale alien traces, like highway overpasses or garbage dumps or other surface modifications. 340,000 images are already publicly available, and by the time the probe has mapped the entire surface, that number should triple.
The other possibility is that there are alien goodies on the moon that we can't see because they're underground in caves or have been buried. If aliens left something behind intentionally, they might have thoughtfully included some sort of transmitter to help us find it, but if we're just looking for old campgrounds, we'll have to get down and dirty on the surface in person.
Of course, the possibility of finding aliens (or their leftovers) on the moon is very, very remote. But on the other hand, the consequences would be so huge that it's probably worth the effort. To get started, click here, and let us know what you find.