If it only takes 600 years to reach Andromeda, and it has lots of species, why didn't the Reapers go there to harvest stuff instead of traveling 50,000 years into dark space?
There should be plenty of galaxies in closer range for them to harvest, shouldn't there?
To be fair, we don't actually know that they don't. We have no reliable information on what the Reapers do in-between harvests; Vigil, the Prothean VI on Ilos, suggests that they go have a nap, but acknowledges that it's only a theory:
Shepard: How do the Reapers survive out in dark space?
Vigil: We have only theories. The researchers here came to believe the Reapers enter prolonged states of inactivity to conserve energy.
It's not impossible that they do spend this time harvesting other galaxies. As far as I know, Bioware hasn't come down hard on this question.
However, if we assume that they don't, the most likely reason has to do with their origins (hinted by Xantec in a comment on the question). For all their bluster, Reapers are a solution to a particular problem, which is discussed in the Leviathan DLC for Mass Effect 3:
Leviathan: Before the cycles, our kind was the apex of life in the galaxy. The lesser species were our thralls, serving our needs. We grew more powerful, and they were cared for. But we could not protect them from themselves. Over time, the species built machines that then destroyed them. Tribute does not flow from a dead race. To solve this problem, we created an intelligence with the mandate to preserve life at any cost.
That Intelligence became the Catalyst, which in turn created the Reapers as its solution. Although not stated explicitly, it's reasonable to assume that the Catalyst was programmed only to care about the Milky Way galaxy; the Leviathans didn't appear interested in "obtaining tribute" from other galaxies, so they would have no reason to seek to preserve life there.
Although the Catalyst (as an artificial intelligence) is theoretically capable of modifying its base programming and broadening the scope of the Harvest, there's no evidence to suggest it has done so; the fact that it continues to pursue its original objective, even after untold millenia, suggests that it hasn't felt the need to self-reflect the way EDI did.
