There is a requirement for backwards compatibility and that is you have to have a hard drive for your Xbox 360. If you bought a Core system you can't play original Xbox games on your system until you pay another $100 for a hard drive. Which is why I suggest not getting a Core and sticking with the Premium system from the start if you can.
The way the backwards compatibility software is updated on your system is done in a couple of different ways. If you have at least an Xbox Live Silver account, the update will download automatically when you try to play a BC game.
Besides it being really convenient to be able to play both Xbox and Xbox 360 games on one system, playing Xbox games on your 360 has other benefits. Each and every compatible Xbox game will be upped to 720p/1080i resolution (assuming you have an HDTV) and will take advantage of full screen anti-aliasing. This makes the games look really, really good. The improvement is actually good enough it almost feels like a whole new experience so it can definitely add new life to your older Xbox games.