Thank you for your post.
FileMaker Pro has the ability to link two different tables together through a common "key" field (or fields). For example, in your "Contact Management" template, there is a field "k_ID_Contact" that links the Contact Management table to the Notes table. In your situation, you would want to connect a Contact to a particular Group. The problem is that one person can belong to many groups, and one group can belong to many people. Therefore, you would need two additional tables. A Groups table, and a "ContactsGroup" table, that combines each. In the Group table, you would have the Group name and a "GroupID" field that is unique for each Group. In the "ContactsGroup" table, you would have a GroupID and k_ID_Contact field so that you can connect to both the Group table and the Contact Management table.
Since you are new to FileMaker Pro, this may take a bit of time to understand and implement. I would take it slowly and look at the "Contact Management" template in greater detail. Pull down the File menu and select "Manage -> Database...", click on the Relationships tab, and you will see a graphical representation of how the tables are linked to one another. This should give you a good starting point.
If you need clarification for any of the above steps, please let me know.