Adding a contact to a group can be done in two ways. You can either open the group and click on the '+ Add member' button at the top of the page and search for the contact you want to add. Or open the contact and click on the '+ Add group' button just below the contact title.
In the second option (via the contact) you can choose to add an existing group or a new group. Press the small cross next to the group if you want to remove the contact from the group.
