You don't have to implement any methods to extend the controller, although obviously it would be rather silly not to. You just inherit from it, and override the methods you wish to alter.
If you don't know what methods to alter, I have to question why you want to extend it?
EDIT:
You may benefit from the two big MVC sample applications, Nerddiner and Music Store. They give you a very good idea of how to make ecommerce sites and the like in MVC. Don't take them as gospel, because they are samples and are intended to be simple. They don't currently make use of best practices like Dependency Injection, or Repository design. Some links to useful tutorials:
Lots of good videos here. The Pluralsight stuff is pretty straight forward
The Music Store tutorial app
NerdDinner tutorial