Depends on the stone. If it's a countertop wetstone, then you want to angle the blade slightly, while you pull and slide outwards. You never want a purely straight pull.
If it's a handheld micro-sharpener, then (while holding the blade at the sharpener designated angle) out and up, minding your fingers and wrist.
If it's a sharpening steel, there's a quite intricate motion that I can't quite describe in words, so I would suggest you visit YouTube.
For more on knife sharpening and diagrams, see: http://www.buckknives.com/about-knives/knife-sharpening/
To clarify on "pull" and "outward":
Pull means to pull the knife directly towards you, as if it were a rope in a tug-of-war game.
Outward means dragging left-to-right or right-to-left, as if drawing a line.
A pull-outward motion would be pulling the knife towards you while dragging it across the stone. This should create a diagonal motion that is heavily accentuated towards having more pulling movement than outward movement.
As noted in the comments, I am an advocate of pulling the sharp edge of the blade with the blunt edge leading, rather than the blade leading. To visualize, imagine a knife in your right hand on a stone. The sharp edge is nearest your left hand, and you move it across the stone towards your right.
Yes. The coarser grit removes material faster, but leaves a rougher finish. The finer side then allows you to bring the edge to a fine polish. The most important thing to remember when sharpening is to keep the blade at a constant angle relative to the stone -- don't change the tilt of the blade as you move or you'll just make a mess of it.
Best Answer
With some experience you can tell by running a finger over the surface.
As a general guide:
grinding stones used for serious repair of knives form damage of long ngelect; will feel disticly rough, not as rough as sand paper but you can definitely feel an abrasive surface.
sharpening stones : these will have a smooth surface but will have a definite 'drag' or friction to them, a bit like rough paper. These are used for normal sharpening of knives.
Polishing/finishing stones : these will feel very smooth with just a hint of texture, a bit like magazine pages. Used for refinement and polishing of a cutting edge .