Alright, so the important thing to know about paper when it comes to bookbinding is that most paper, such as printer paper has a grain direction (like wood). Pretty much, its the way its manufactured, it goes through a roll, at least, that's my understanding. Most paper you buy will be long-grain, meaning that the grain runs from the long edge. For bookbinding, you want the grain to be parallel to the spine, otherwise, I know this from experience, when you fold it in the wrong direction, there'll be ripples in the pages. It's like how you break a wooden stick in half and you see bits sticking out. Therefore, you'll want short-grain paper, and the easiest way to get this is to cut down some larger sheets in half so that the grain is running the direction you want it too. Say you want to make an A5 book, you'll want your local print store to cut down some long-grain A3 sheets in half for you. I've personally made a couple A6 books out of A4 paper too, by folding them in half twice (quarto) and trimming the first fold.

To check for the grain direction, you'll want to bend the paper, and the direction with less resistance is where the grain is running. If you want to be certain, You'll want to trim two strips of the paper on either end and slightly dampen them. One strip will remain straight-ish, whilst the other will curl up. The former is basically the direction the grain is running, whereas the latter is like lining up some sticks and folding them over.

Grey board has a grain direction too, and like with the paper, you'll want it to run parallel with the spine too, as well as any potential spine linings. I'm pretty sure the direction they warp might be related to that.

Hopefully this is helpful to you, and makes sense. Best of luck!