As I think you've gathered, a water wheel doesn't shuffle water, but is used to power things (like screw pumps).
If you only want to move a reasonably small amount of water the "bucket brigade" is the easiest way. For larger amounts or a more permanent setup a water movement solution is better, and they generally either make use of natural flow (sideways/downwards) or use screw pumps followed by natural flow.
I'd read the wiki page on the pump stack, reread it, and then implement a stack reading the wiki page carefully along the way (at least that's what I did). You need at least a single tile "cistern" for each level of the pump stack, since the lower one needs somewhere to output the water, and the one above needs somewhere to fetch water from. The larger 3 tile "cisterns" that may be mentioned are not needed for water: they're there to reduce the temperature calculation burden for magma pumping.
Screw pumps are powered through a gear or an axle going into the impassable tile of the pump, or an adjacent machine (such as another screw pump). Typically a pump stack is thus powered through the topmost pump from above. It's possible to power a pump stack from the side, but that causes a leakage (although I've seen it said you can obsidianize the power conduit and it will still work).
There are a number of security issues with pump stacks: since pumps are buildings, building destroyers can deconstruct them. The power train is also a possible entry for nasties, in particular flying ones. It's possible to build safe power trains and to protect pump stacks, but it requires a bit of work, and an exposed water wheel may very well have to be reconstructed after a gobbo siege containing trolls (or ogres), so don't have a setup where the fortress is flooded if you lose power...
Water wheels have to hang from something, which typically is a gear or a horizontal axle beside the center of the wheel. You can also hang a water wheel from another, supported, water wheel, so you can span your river with a thick roll of water wheels hanging from a single gear/axle. Since machinery conduct power, all the power from all these wheels is sent through you power train.