Traditional Arkansas sharpening stones are natural quarried stones. They are graded by hardness (density) rather than grit and are finer than manmade sharpening stones. They are known as oilstones because you need to use oil when sharpening tools with them. The oil is called honing oil and is usually a light mineral or petroleum based oil. Cooking oil is not recommended. The oil acts as a lubricant and suspends the steel shavings, preventing clogging. I use Wahl oil which is a lubricating and cleaning oil for my dog clippers.
My Arkansas stones have different shaped edges which are useful for the internal edges of the U and V gouges. I use Swiss made Pfeil woodworkers gouges to cut my lino plates.
First, move the blade of the U gouge up and down the stone in a circular movement (red arrow on picture). If this was all you did then only a small part of the U shaped blade would stay in contact with the stone and get sharpened.
In order to sharpen the whole U shaped profile of the blade you also need to slowly rock the blade from side to side (blue arrows on picture), at the same time as you are going up and down the stone in a circular motion. It sounds like trying to pat your head and rub your tummy at the same time, but it’s not as bad as it sounds, once you give it a go! Keep at this until the blade is nice and sharp then use the curved edge of the stone to remove any burrs on the inside edge of the gouge.
Finally clean your stones with a little more oil by rubbing it in a circular motion with a rag across the surface of the stone. The small metal particles need to be removed from the stone or else they will clog it up. Then wipe with some clean rag or kitchen paper. Rinse the sone under running water, then dry thoroughly.
I hope this helps keep your tools nice and sharp. Happy cutting.