Space is big. The distances involved are ridiculous, the energies are ludicrous, the costs are somethingelseous. The New Horizons probe (due to reach Pluto in 2015) is the fastest vehicle ever created. At its top speed of 16.26 km/s (36,400 mph) New Horizons would take over 80,000 years to reach Alpha Centauri (the nearest star system).
(The following paragraph is wrong. There’s a redaction here: My bad: Have aliens ever visited Earth?)
A quick calculation shows that the fastest that a ship can reasonably travel, using its own fuel, is about 11% of the speed of light (0.11C). This calculation assumes that the ship is essentially all hydrogen tanks, that all of this hydrogen is fused into helium, and that all of the resulting energy is converted into thrust. However, even at this speed, it would take about 40 years to get to Alpha Centauri. Any realistic method used to get from one star system to another involves millennia of travel time.
(This last paragraph was wrong.)
Adding to that, beyond exploration, there aren’t any good reasons to leave your home solar system. The resources you’re likely to find are about the same as the resources you’d find in your home system (nearby stars tend to form in regions with similar chemical make ups), while the costs of getting there are amazingly high.
It has been proposed that aliens may have a technology that would allow them to travel faster than light (FTL). Sadly, there’s a lot wrong with that. Unlike the sound barrier, which was an engineering problem, the light barrier is written into the universe. In fact, based on everything we know about time and distance today (I’m talking relativity here), the question of FTL travel doesn’t even make logical sense. However, if you’re interested in technology based on unknown (contradictory) science, then there are plenty of experts to talk to. Scientists almost never tell you what you want to hear.
On a personal note, if I made the trip to another star system and found intelligent life, I would spend the rest of my life there telling everybody about it and bragging to the native population. If aliens have been to Earth, they’ve been suspiciously cool about it.
Mathematician: I disagree with the physicist’s answer in one significant way. While it’s true that the distances in our universes are insanely large, and that the speed of travel is capped by the speed of light, the theory of relativity can actually step in to save the day (when it comes to aliens traveling enormous distances).
If aliens were able get up to speeds approaching that of light (using, say, a Bussard ramjet) then they would experience a twin paradox type situation (essentially, time would slow down on their ship compared to the passage of time on Earth). Hence, while a trip from Alpha Centauri to Earth (traveling at almost the speed of light) would take something like 4.5 years from the point of view of us earthlings, to the aliens such a journey could take very little time (thank you time dilation and length contraction). The closer they got to the speed of light, the less time would elapse (for them) on their journey (and the more energy such a trip would require, with an infinite amount of energy being necessary to travel exactly at light speed). If they ever did travel at exactly the speed of light (which is almost certainly impossible) the entire trip would take exactly 0 time from the viewpoint of the spaceship. Due to these relativistic effects, Aliens could theoretically travel enormous interstellar distances while experiencing very little aging. The members of their species on their home planet(s), however, would continue aging at the usual pace, so long journeys (of, say, thousands of light years, which is still not much compared to the hundred thousand light year diameter of the milky way galaxy) would presumably lead to the death of all of one’s planet bound acquaintances. Therefore, such a journey would only make sense for an explorer who had no intentions of ever returning home, or for large star ships that acted as their own colonies. Perhaps it’s also worth mentioning that when aliens actually arrived at Earth after such a trip, our civilization would have changed dramatically from the time they departed. See here for an article that discusses some of these questions in greater detail.