In Windows, there was a little section of the start menu bar next to the clock that had notifications and other programs that showed up in it (Like Windows Updates). I set a bar like that to appear in Ubuntu, but I notice that I can't see my firewall (Firestarter) or my bittorrent client (Azureus) in the notifications tray. I don't know if my firewall is running, but Azureus still downloads even after I close it, so I'm pretty sure it's still running somewhere.

Is there a way to make things like Azureus and Firestarter appear in that area?

You need to right click in an empty area of a panel and select "Add to Panel." For example that how I got my network monitor icon in the panel. I have no experience with bittorrent in a gnome environment.

Yeah, I have the notification area in the panel, but things like Azureus don't show up. Amarok and FrostWire do, but nothing else seems to.

Firestarter doesn't have a tray icon; you can check it's running with gnome-system-monitor or ps -e. Azureus's tray icon is broken currently, as far as I know.