Staining will highlight any imperfections in the wood. From the photo it looks like something has been wiped across the grain. That could be when you wiped the sawdust away or possibly if you wiped the stain across the grain.
Having a good surface to stain is pretty much essential as the stain will highlight even the smallest imperfections.
Applying thin coats as John suggests is the best way.
Also, don't forget that the wood will age - especially for something left outside - so any imperfections you see now will blend over time.
There are two critical steps in prepping kitchen walls.
Clean and degloss
Cleaning is critical: TSP or TSP substitute.
Deglossing: either light sanding (120 grit) or a chemical deglosser
Once a surface is deglossed, it should be primed, especially if going from one paint type to another.
A deglossed oil undercoat can be primed with a latex primer. Either oil or latex can then be used as a topcoat.
A deglossed oil undercoat can be primed with a oil primer. Either oil or latex can then be used as a topcoat.
It is better to stick within families of paint types, if you can, but going back and forth between types is possible with primers.
At this stage in coating, I'd let everything dry for a week, then sand lightly. If the previous layers are adhering, then proceed with latex primer, latex topcoat.
If,, however, they are sloughing or chipping or you can easily pull a layer off if you leave painters tape on overnight (and then pull it off), then you must sand down to the last well adhered layer and then repaint. Good luck.
Best Answer
After staining give a coat of clear finish, sanding it with a fine sandpaper (400G) to remove any raised grain or any dust that may have settled onto the drying finish. This will give you a good base to add your paint and the sanded surface will give it good adhesion. Steel wool has its purpose, but I would not use it here. That's only my opinion. My experience is the steel wool sheds metal bits everywhere, and may be a deal to clean up completely.
After the recommended dry time for the paint is over, give it your final clear coat. I know this did not answer your question about how long to wait before the top coat, but depending on what brand of paint you use, makes a difference. Each paint container always list how long it needs to dry before the next coat. Many usually only need 8 hours.
If you do not have that, the safe rule of thumb is, 24 hours if oil over oil, 2 days if water over oil. I don't know what to recommend if oil over water.