Getting rid of water on the lower levels

Well you can't tunnel off the side of the map because the edge tile is never diggable, so that's out anyway. So it turns out you can turn an edge tile into fortifications which will drain water. First smooth it, then carve fortifications. (Thanks Nakedible!)

However, for those who consider this cheating or who are using a non-infinite water source, the only way is, indeed, a pump stack. This, while time consuming, isn't as terrible as you might think. (You'll probably have to do this later to bring magma up from the deeps, anyway.) The wiki has a description of optimal pump stack construction.

This will allow you to build a stack of mechanically operated pumps without putting extra machinery on each level. You only have to power one of the pumps in the stack. (Assuming that power source provides enough power for the whole stack.) Combine this with the (somewhat cheating) dwarven water-wheel reactor and you have a relatively painless pumping solution.

I like to put two gears between the power source and the pump stack and attach a lever to the gear nearest the pump stack. This allows me to turn the whole thing off if chaos ensues for any reason or I need to do maintenance to the system.

The alternative is of course a genuine water wheel setup or windmill setups, but these options will force you to be more choosy about where you locate the construction. Any axles or gears that lead from your power source to the pump stack consume power!

You might want to try building a smaller pump stack before going for one that spans 50 levels or more. Keep in mind it's very easy to flood your fortress with Fun!

From what I understand, if you divert the water into a pre-existing infinite water source underground (such as lakes in caverns and aquifers) the water will basically be eaten as fast as is can path in. This isn't as much "cheating" because these natural sources of water are often already open at the map edges, so you can think of the water as flowing off the map rather than disappearing into source blocks of water. Good luck! (And I would say "have fun!" but that's more of a curse than encouragement in this game... :P)

I tried doing the same thing but I drained it into the first cavern levels without any problems other than flooding part of the caverns. I seem to recall that there was water leaving the edge of the map in one part of the caverns that provided convenient drainage.