.NET Standard is a target for libraries and can be used with any of the .NET runtimes: .NET Framework, .NET Core, Mono or even Xamarin for mobile. If you have a common library with all the game logic, then I recommend you target .NET Standard with that project.
For your per platform projects (you need 1 for desktop and two more for Android and iOS) the target depends on how you want to distribute your game. For mobile you need to use the Xamarin toolchain. For desktop things are a bit more complicated. On Windows users have .NET Framework already installed, so you can distribute your game without a .NET runtime by targeting .NET Framework on Windows. The .NET version will determine what windows versions your game will run on. I think .NET 4.0 runs as far back as XP and up while 4.5 runs on Vista and up (but not 100% sure about that). Now, you can include a runtime with your game as well, but your game will be a little bigger.
For Linux and Mac, there's no runtime that comes with the OS, so typically a .NET runtime is bundled with the game so users don't have to install one and can potentially run into compatibility issues. That runtime used to be Mono, but with .NET Core things have become a lot easier. With a build flag you can build your game into a single executable that includes the runtime for the platform of your choice. So I recommend you go with .NET Core for Linux and Mac. If you don't care as much about the size of your game, you can also use this method to build your game for Windows. It saves I think something like 20-30MB.