Greetings Silentpanda613,
Being able to download movies and shows from the iTunes Store when connected to Wi-Fi is important, and we’re here to help. We’re glad you were able to test different Wi-Fi networks, and isolate the issue to your home Wi-Fi. Since you've isolated the issue, let's begin with the steps in the following article.
If your iPhone, iPad, or iPod touch won’t connect to a Wi-Fi network
While the title of the article references not being able to connect to a network at all, these steps can be helpful if you’re connected, but unable to use data enabled features. If you continue to have issues after those steps, try the steps outlined in the article below.
If you can’t connect to the App Store, iTunes Store, or other Apple services
Let us know how that goes!