Joining a group from an invitation:
Go to your Profile page
Tap the envelope icon "Invites" button at the top of the page. (If you do not see one, you do not have any invites)
Tap accept for the group you wish to join.
Requesting to be part of a group:
Go to the Leaderboard page and select the Groups leaderboard.
Tap on a group you wish to join.
Tap on the "Apply To Group" button.
Wait for your request to be accepted.