So how would I use this on Linux? I'm seeing different versions of VS to download from the official website. Only VS Code is available on Linux?! Does/will that contain Xamarin?!

Visual Studio proper is still a Windows only product. Microsoft's aim here is to make Windows and Visual Studio the only solution you'll ever need to develop for every platform under the sun, be it iOS, Mac OS, Android, Linux, Windows, or web.

Visual Studio Code is a very lightweight semi-open source IDE for writing code, and not much else.