Apple TV Plus

Trouble signing on to apple tv plus on windows pc - I don't have an apple device, can this be done?

Yes, it can be done. Not the easiest, though.

First get a properly configured Apple ID account, with full user details and email confirmation (respond to test message) and payment method.

How to create a new Apple ID - Apple Support, if you don’t have one yet.

Account management is best done in the iTunes application.

Then you may subscribe. Get Apple TV+ - Apple Support

No matter what I do, even after I'm logged in with my Apple ID, it never allows me to sign up. It spins for infinity on the sign in screen after it successfully recognizes my ID and my password.

The Apple Support staff would be able to check your account and assist you in getting started. . Get support - Apple Support