I've seen neither the Clone Wars TV show nor the movie, so I'm at a loss for what is happening in most of the cutscenes in Lego Star Wars 3. If I wanted to understand them, would I need to watch both the shows and the movie, or is the game based on just one of them?
The game is based on the Clone Wars movie, seasons 1 & 2 of the series and a bit of the Attack of the Clones film.
As with other similar LEGO games, knowing the source material isn't necessary to understand the game, even if you do know the story, you'll see the games like to make liberal interpretations of it. In fact, the first Star Wars LEGO game, which was based on Episodes I-III, came out before Ep. III was even in theatres. I don't think LucasFilm would've allowed that if they thought it will spoil the film in any way.