To add a member to a group, you first need to go into the group membership package and select the 'group' that you wish to add the member to.
Once in the group's membership profile, click on the 'Member Info' tab, this will allow you to view all the members within that group. At the bottom of the list of members in that group, you will see 'Add Member'. Click on that button to add a new member to that group.
Here you can add a new or existing contact to the group. Once you're happy with all the details, make sure you use the 'Save' button at the bottom of the page to add the member to the group.
You can also remove members from a group using the 'Remove' button adjacent to each member in the 'Member Info' tab.