There are several ways to start this, that lead to the same sign-up. This is probably the easiest:
Get Apple TV+ - Apple Support
The blue button on this ↑ page automatically redirects to the TV app on your current model iPhone (or iPad, or Mac). It would open the TV app and show a sign-up option there for Apple TV+. Follow the on-screen instructions. Once you have started the subscription, the content will be available on any of your devices that has the Apple TV app, when signing in to the app with the matching Apple ID account.
We are using YouTube TV
That is completely separate from Apple TV+. Apple TV content will be available through the Apple TV app on any of your devices, and not through the YouTube TV app.
On what TV or TV-connected media player do you have the YouTube TV app and other smart apps? You will want to get the Apple TV app on that too.
Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support
Watch on these compatible smart TVs and streaming devices - Apple Support
How to watch Apple TV+, MLS Season Pass, and more - Apple Support