I wrote a lot of C# code in Visual Studio for Windows, but just recently started using Visual Studio Code on Linux for C# projects. Here are some steps how I made it working.
I assume you have working Visual Studio Code and you already installed .net core 3.1. Take you time to make it working before continuing.
The only extension you need is the C# for Visual Studio Code.
There is no magic button to add new solution and its project, but adding it from shell is also very convenient.
Enable build and debug
When you first time open the directory you will be prompted to add missing assets. It will allow you to run (CTRL-F5) or debug (F5) your code.