What you have there looks like oak plywood.
Stain is meant to be applied to bare wood. If the wood is already clear-coated, the stain won't be able to soak into the surface. So you'd have to sand it thoroughly first to take off the existing finish. But the problem with sanding plywood is that the veneer is usually very thin, so there's a high risk that you'll go through the veneer completely if you sand too much. So, I would recommend paint, not stain.
If anybody claims that there's no surface prep required, then you know they're a goddamn liar. Prepping the surface is the most important part of any painting project. At the very least, you'll need to scrub the doors with soap or degreaser to remove the airborne kitchen funk that's been accumulating over the years. If there are any chips or holes, fill them with wood filler, then sand the filler flat (sand by hand, and don't overdo it - see previous paragraph). Then lightly sand the whole surface to scuff up the finish, this helps the paint adhere. Plan on doing one coat of primer, two coats of paint.
Lastly, don't buy paint from Amazon - go to a local paint store or hardware store. They'll be able to mix custom colors for you, and offer technical advice.