The Lord begins His first official covenant with Noah. After the horrors of the global flood subsided, humanity would have felt understandable anxiety over what God might do in the future. Noah and his family needed reassurance that, no matter how geography or meteorology might have changed, normal life could go on without men constantly worrying about death suddenly falling down from the skies. Noah sacrifices clean animals to the Lord (and as a side-note, it is interesting that Noah knew there was a difference between clean and unclean animals, centuries before Moses wrote down the dietary laws).
The Lord responds to Noah with a promise, Never again would the Lord destroy all life with a flood. He would make sure that the atmospheric seasons would endure in predictable, reliable fashion. Weather would be generally predictable. Human life could go forward. "Weatherman" could be a career. God's earlier reduction of the average individual human life-span would also curtail the degree to which human evil could flourish.
This Noahic covenant is a promise that extends to all of us. It is one of the reasons I do not fear global warming (now re-named global climate-change, since man-made global warming has been shown to be an uncontrollable function of nature, and a politically-motivated fabrication). God does not intend to annihilate the race, and pledged not to inflict world-wide catastrophe by water ever again. He has your whole world in His hands.
Rain didn't exist in the Edenic ecology. Mists rose from the ground during the day, then settled back down as dew in the night. Rain would have terrified the human race. So God appoints the rainbow to remind us of God's mercy. After judgment comes life and peace -- a nature-picture of the cross and the resurrection. One more thing not to worry about.