Does Kindle charge for every month?

No. You can download the app for free, and some kindle books are free. I mostly use it for philosophy and history texts. It's the 100+ year old translations I get free. But if you are into contemporary non-fiction you can get that sometimes free if the author is trying to get word of mouth going.

The Kindle Fire tablet costs money. The Amazon Prime Video app costs money too.

I believe so, $9.99 after a three month trial.