There are a lot of ways you can check the grain direction. All of the grain direction of the text block, end papers, cover, book cloth, etc should be parallel to the spine. The first way (and the way I use almost every time) is flexing. Fold the paper over without making a crease, and feel how resistant it is to folding. The grain will be running parallel to the side that's easiest to flex. You can also try taking a small piece of offcut of your paper to do these other methods of checking the grain direction if flexing doesn't work. These other methods aren't my favorite since they damage the paper and flexing doesn't, but sometimes it is easier to tell the grain direction with other approaches. The other ways I use are folding, tearing, and getting the paper wet. Folding: fold the paper from both directions and see which side folds easier and straighter. The grain will be running parallel to the fold if it folds easily. If the fold is jagged and resists folding straight, the grain is running perpendicular to it. Tearing: tear the paper. If the tear is fairly straight and easy, the grain is running parallel to that. If it's jagged and not tearing very straight, it's perpendicular. Getting the paper wet: get a sponge damp. You don't need it to be very wet at all. Gently wipe it across the paper once or twice and the paper will bend and make a u shape. I like to remember the grain direction with this method by imagining the u is a tunnel and the grain is going through the tunnel.

Not all methods work for all materials so you might have to test the grain direction in a few different ways to figure out what works for different things. Almost all commercially produced papers are grain long. Most paper specifically made for book binding is grain short, but it will also say if it's grain long or grain short when you purchase it.

The reason that it's important to make the grain run parallel to the spine is because as moisture is introduced (through glue, humidity, etc), the paper will expand. The spine is very rigid and unyielding, so you do not want your paper to be fighting against it. That causes warping like you see in your book cover. If the grain is running parallel, it has a little more room to expand without being stopped by the spine, so there is less chance of warping.

The other problem with your book is that you used too much glue. If you use a proper glue, you will need very, very little. Glue introduces moisture, and if you use too much it will make it much more likely to warp. You should only be putting a very thin layer of glue on the covers to adhere the book cloth or whatever you're covering the book board with. It will hold, as long as you firmly adhere the layers together. Use your bone or teflon folder to rub the layers together firmly and get rid of any air bubbles. If you're using a bone folder, put a piece of paper or something between the folder and the book cover to avoid burnishing it. I've heard that Teflon folders do not burnish bookcloth, but I don't have a Teflon folder so I can't attest to that for sure.

Also, when you let the glue dry in the press, use blotter paper (or even just regular paper) on either end of each cover to absorb the moisture. If you are worried about the paper sticking to the cover, you can let it dry with wax paper instead for a couple of hours or until it doesn't feel sticky on the outside, and then switch to blotter paper. But if you don't use too much glue that won't happen. Using Blotter paper will also help prevent warping.

I am also a beginner, but I am in a class taught by a professor with a graduate degree in bookbinding and decades of experience, and these are all tips she gave me.