I've played df for over 10 years now (to and fro) and I've never bothered to learn how to deal with aquifers. These days I just remove them from the game entirely because they seem to be absolutely everywhere in the modern versions. I don't remember them being so ubiquitous in the olden days.
It's been a while since I last played (FPS death concerns and all... even though I have the beefiest PCs I've had in a while), but there were basically three methods
- Find a breach in the aquifer. More likely than not the aquifer doesn't fill the whole layer, and often you can find areas in which there is no aquifer and you can dig down to the lower levels. This is likely the simplest method but depending on how lucky you are and the layout of the strata, it might be either very easy, or very hard (and annoying: "digging designation cancelled: dampstone located". Not sure whether that warning can be turned off without your dwarves flooding your fortress) or impossible.
- Collapse natural terrain on top: This is favored by some people. I personally found it cumbersome. The theory goes that if you collapse enough natural walls on top of the aquifer, they will replace the aquifer tiles, and you'll be able to dig through. Artificial walls do NOT replace the underlying terrain so it doesn't work. Anyway, I never liked this one because it requires you to have sufficient overhead Z levels to begin with, setup is very hard, and there are no guarantees (you don't know how DEEP the aquifer layer is)
- Pump rig: I used to favor this one. Basically you use a pump to remove water from an aquifer tile onto another aquifer tile. Add pumps and dig more tiles as necessary. The main problem is that in recent game versions aquifer tiles are more often than not dirt, rather than solid rock. Which is a problem because smoothing rock to remove the aquifer is simple, whereas with dirt you have to dig it out and replace it with an artificial wall, whose construction is being interrupted over and over... you get the idea. All in all, it's pretty foolproof, but very labor intensive. And if there are multiple layers you'll need bigger overhead holes as to be able to repeat the process, which might require you to reform your original hole, making it even more labor intensive, etc..
My biggest qualm is that there is not a simple automated way of doing it, and there really should be one. Eg, making building jobs autorestart, etc...
All that being said, I do like aquifers as infinite sources of water to play around.