See also the official documentation:
Help Center > Members Manual > Understanding Groups.io accounts > Logging in to your account

How to get online at Groups.io the first time

At Groups.io using your browser on ANY internet enabled device with a browser:

Note: "Click" can mean "Tap" on personal devices

1. Use this link in your browser  https://groups.io/login
2. On that page click on the button that says "Email me a link to log in."
3. When you get the email on your device, click on the link or copy and paste it into your Internet browser and go there. You are officially now "IN." 
4. The first thing you should do is to Bookmark that page so you don't have to go through all of the above again.
5. The second thing is to set up/change a password for the same reason:
   • Click on your identity in the upper right of your screen.
   • Click on Profile in the dropdown.
   • In the box next to the word Password, enter a new password.
   • Click on the Change Password button.
6. Near the top of the screen you should see something like "Your Groups," which you should click on.
7. Click on the name of your group.
8. You should now be at the Messages.
9. Bookmark this page on your device.

That should work for anyone on any device.