How do you search for movies or series on apple tv ?

On the website tv.apple.com, just the Apple TV+ content is presented, and that is all that is available there. Apple TV+ is the streaming service that brings Apple Originals. The website currently does not have a text search function. But all its content is featured on the main page.

In the Apple TV app on streaming devices, you have access to much more content (not just Apple TV+), including iTunes Store movies and TV shows for buy or rent, and premium Apple TV app Channels to subscribe to. The Apple TV app on streaming devices does feature Search functionality.