What do I do if I cannot get the shows to play on my Laptop. I have had this issue for a few days now. I have tried turning it off and on, changing passwords and I am up to date with payments.

MacBook Pro 2013

There were a lot of 2013 models MacBook Pro, with different capabilities, and different maximum versions macOS.

Identify your MacBook Pro model - Apple Support

Verify that you have the latest supported version macOS for your model (Update macOS on Mac - Apple Support). What is this version macOS?

when I try to watch shows on Apple TV+, I get the error message "browser update needed."

Safari is bundled with macOS, and therefore the versions are linked. The latest version Safari that is available for this model computer is determined by the latest version macOS that it supports and has installed. An old version of Safari may not be enough. Consider Chrome or Firefox browsers, as those typically have a bit more recent version available with support for later web technologies. A bit ironic to get better service through a third party product, but c’est la vie. But even those browser makers abandon older versions macOS at some point.

What kind of laptop? Mac or Windows PC? What OS version?

Are you watching in a browser (up-to-date Chrome or Firefox), on the site tv.apple.com?

Did you sign in to the website, through the blue Sign In button in the upper right corner, using your Apple ID account that has the Apple TV+ subscription linked to it?

What happens instead, when a selected show does not play? Do you get an error message? What does that say exactly?

I am experiencing this also. MacBook Pro 2013, when I ty to watch shows on Apple TV+, I get the error message "browser update needed." Yet no browser update is listed anywhere.