It is a motion or movement that you are trying to describe. A move connotes some very specific movement that people might have to practice - like a dance move or a fighting move. Move is used more often as a verb than a noun. Motion or movement have different dialect-dependent connotations; in some parts of the UK, referring to a motion has a particular toilet-related meaning. Actually, so does movement, but it's much less common - so there's a lower chance of anyone snickering at movement. Both nouns can be abstract and uncountable, referring to movement in general, or countable to refer to a particular instance or type of movement. Their usage is not entirely equivalent, however - you can describe an object as being in motion, but not in movement.
Turn from and turn away from have essentially the same meaning, but turn from tends to be used more poetically or metaphorically:
You must turn from your wicked ways and find salvation.
Both expressions can be used for the physical movement or the metaphorical movement (as you quoted in comment, "to refuse to accept or to use something"), but in my experience turn away from is more often physical, and turn from more often metaphorical. This is just using the word turn but using away to indicate direction.
Of course, you could just say "turn around" and have done with it - if everyone knows where he was facing, then obviously he's turning away from that direction.
You can use a lot of verbs for the movement towards the cooker. If you use step, I would use step up to, though I can't put my finger on why. Step to is of course an expression on its own, probably an abbreviation of "step to it", essentially meaning "get on with it". That might be why I prefer "step up to". You can also use move (up) to; they mean the same thing, but using step implies that he is walking, and doesn't have to move far. Come up to is also reasonable, though overall I'd prefer step up to in this situation.