Hello, I thought I'd answer this as well! I'm not sure about the availability of Finnish movies and tv shows, but I'd like to give you a few recommendations anyway.
There's this show called Pasila that's pretty popular in Finland, it's a sort of adults' cartoon about the cops in Helsinki and very humorous even though the language may be a bit difficult for beginners.
Children's cartoons might be a good idea as well since they tend to have simpler vocabulary (for example, I can understand Moomins in Japanese). Youtube seems to have many cartoons that are dubbed in Finnish, for example Pokemon and Teenage Mutant Ninja Turtles.
In case you use Netflix it might also be a good idea to get Hola and try accessing the Finnish Netflix? Just a thought.
As for movies, one of my favourite Finnish movies is Napapiirin Sankarit (Lapland Odyssey) which my American friend also loves. Another good one is Star Wreck: In the Pirkinning, a low budget scifi parody that at least used to be available for free online.