Yeah, movements along the curve are caused by situations where the price and quantity AREN'T in equilibrium (so essentially, directly after a shift in either the demand and supply curve has occurred), so the price/quantity will move ALONG either the supply or demand curve until it reaches the equilibrium point.
The 3 key economic decisions are what, how and for whom to produce?
The price system determines what to produce because firms will look to produce things that have a high relative price to obtain more profit.
The decision about how to produce is left to firms who want to maximise profit potential (I guess that falls under the umbrella of 'the price system'), who will produce goods and services in the most efficient/cost effective way in order to cut costs/maximise profits (because they have a direct incentive to - compared to, say, socialism where firms have bugger all reason to be efficient cause they'll get paid the same regardless from the government).
The decision about for whom to produce is answered by the price system because the people who have the greatest willingness to pay for goods and services are the ones who end up getting them (they're the ones who pay the price).