Upward-moving air cools, and if this cooling is great enough, the air will cool to the point where it cannot hold any more water vapor. This water vapor then condenses, and falls as rain or snow. It depends upon where they start. An air mass that sits a long time over a hot, wet ocean will become hot and wet. An air mass that sits a long time over a cold, dry place will become cold and dry.
Weather can move in any direction.
In the United States, weather most often moves west to east, because that is the direction the jet stream blows. The jet stream carries weather systems along with it. The answer is complicated, but it happens as a result of the way the sun heats the equator more than the poles, and the speed with which the earth spins. There has been a lot of research done on the subject by many scientists. Scientists capture snowflakes and photograph them to study their patterns.