Product keys are used to add games to your library, but these keys are only produced when supplied by third parties or the software developers themselves. If you buy a game directly on Steam, you will never see a product key unless the game has its own key system separate from Steam. (Some games do this to lock in-game content that they don't want to manage through Steam's existing DLC system.)
Once a game is in your Steam library, you can download it and play it. You need only have Steam running in order to play the game once it has been downloaded. Steam will also keep it up-to-date for you.
