The "art" plywood is baltic birch and comes in packs of 5 cut to size. The advantage of this material is that the cuts are perfectly clean and the board should be very square. The downside is that they are only available in 1/8" thickness. This means that sizes are relatively limited without cradling. In fact, I will only use this material up to 12X16. That may actually be the upper size limit available, I don't remember offhand. One of my favorite things about painting on board is that it is extremely easy to frame these paintings with only a point driver. Having to resort to a cradle for large sizes eliminates this ease and also adds a lot of weight to the finished piece!
For larger pieces, I go with good old hardware store sheet plywood. I usually use 1/4" oak panels. It is also available in maple and I think birch may be available as well. The 1/4" thickness resists warping in sizes up to 30X30. I have found that propensity to warp is a function of the individual sheet more than anything and is probably due to insufficient curing at the factory or a defect in the internal plies. I have had some that warp up badly and some that stay nice and flat no matter how or where stored. If you buy a sheet that wants to warp, you will be forced to cut smaller sizes. But I'd say in more than 90% of the panels I've bought, I've had no issues with warping at all. A good stable sheet might could be cut into sizes larger than 30" but only if used in a fairly square aspect ratio. The longer the aspect ratio, the more likely you are to see some warpage.
After cutting my panels to size, I use coarse sandpaper to remove burrs and smooth the cut edges. At this stage, you will have two edges that cannot be smoothed very well. These edges are cut against the grain of the internal plies and will look porous. I'll fix that later. I will also lightly sand both faces. You'll notice that one face is slightly smoother than the other. That's the face I usually paint on but not always. Sometimes I might find a flaw in the smooth face or maybe I want that rougher surface for some reason. After wiping the faces down, it's time for gesso. Using the precut baltic birch, I avoid all of the previous steps. I use acrylic dispersion "gesso" from Liquitex, the relatively thin stuff. I use 3 coats on the face I intend to paint on and usually one or two on the back side. Whether or not I gesso the backside is sometimes dependent upon time and how impatient I am to use that size. I do not sand between coats unless the panel is small and I have large brushmarks left. I have learned that the brushmarks usually are aesthetically pleasing to me in large sizes but distracting in smaller ones.
I know that some artists use a primer or sealer but I do not and I have a very specific reason for that. Most artists use what's called GAC100 from Gamblin as a primer. I also know of at least one artist who uses one coat of clear gesso first and then 2-3 coats of the standard gesso over that. Remember now, when I say gesso, I am referring to acrylic dispersion primer and not traditional, real gesso.
On a hot day here in Arkansas, I can do all 3 coats in a single day. The longer, more difficult part is sealing the edges. This is not a necessary step but I like to do it. It produces a more professional product I think. I will gesso all 4 cut edges and sand them as smooth as possible. It may take several coats for the rough edges cut against the grain of the internal plies. I still use only coarse grain sandpaper. I don't think you need to use anything very fine, all you want is a sealed edge with no burrs. This step is something you shouldn't bother with if you are using the precut 1/8" panels. With these panels, they are so thin you will only succeed in building up a bead of gesso on the face of the panel and it won't do anything for the cut edge.