To add a new discussion to a group, you must first have joined a group.  To see if you are in any groups, click on the My Groups link underneath your profile picture on the home page.  If you are not in any groups, you can either Join a Group or Create a New Group. 

To add a discussion, choose the group that you are updating under "My Groups."  Once on the group's page, scroll down to the Discussion section.  Here you can see existing discussions, which you can comment on, or Create Discussion by clicking on the link at the bottom of the Discussion box.  From there, you can title your discussion and create the first message that you want to be posted so others may comment.

To Post on the Group's Wall, return to the chosen group's home page.  Scroll down to the Wall box.  A box should be there for you to write your message in to post on the group's wall.  Once you type your message, click the "Add a Wall Post" button and your post will appear directly below the message box along with your profile picture.