NOTE: Microsoft stopped updating their backwards compatibility for original Xbox games years ago.
You'd just need to connect the system online, go through making a Silver account (free), start up the game and then download any current updates.
In either version, you get firmware updates for your hardware to increase performance or update certain software features.
With Xbox Live, when you sign in using your Gamertag, you are automatically notified of any new updates. How do you update your Xbox 360 without an Internet connection or a very slow Internet connection?

But since it downloaded the system updates and confirmed the XBox Live account, I figure that connectivity is not the problem.
However, I can't seem to get it to update any of the games.
As in, there's no notification that an update is available, even after starting a game.
Like all game consoles, there are occasional updates to add new features such as social networking tools like Facebook, Twitter, etc or the ability to play different types of files.
Xbox Live has two types of offerings, free and Gold Membership.