I take it you are successfully signed in now?
… said that Apple TV was free for a month, then $4.99 a month.
I think that would be a message for the Apple TV+ streaming service, featuring Apple Originals.
It would have a week long free trial period, not a month.
It would have a year free trial, if you purchase a new qualifying Apple device.
I never saw options like the Tom Hanks movie, Greyhound, or other shows
Hmm, ‘Greyhound’ is part of Apple TV+, the subscription service mentioned above. You would access that through the Apple TV app.

Verify that your Apple TV software is up-to-date. How to update your Apple TV - Apple Support
Not all that important, I can probably give it away to someone more knowledgeable than this 78-year-old.
You came here for assistance. And we would like to provide that, if you still want us to do so.