Read the section from the Swing tutorial on How to Use Menus. It has a working example that will show you how to create sub menus.
But for me to send the int-values of each JMenuItem to an ActionListener I would need 3 different ActionListeners. And they would need to be hardcoded.
Yes, that is the way is should be done. You should have 3 separate Actions. Each Action should have a value to specify the font size.
You can also read Text Component Features for a working example of a simple editor that uses the default editor kit Actions to do the customization of the text.
The tutorial is a good place to start for all Swing basics.