IIS Express is an extension that will basically run your project folder as a website. It's an extremely handy tool that I couldn't live without. Usage is simple, and noted below. I should also point out that IIS Express is only available for Windows and will not work on Linux or OSX.
- File > Open Folder
- Select the folder with your PHP and HTML within
- Install the IIS Extension from the extension menu. (Or with the command palette using "ext install iis-express")
- Open the command palette (CTRL + SHIFT + P)
- Type "IIS Express: Run Website"
- This starts the web server and will open a browser and connect to your website for testing.
- To stop the server, simply type the command a second time.
More information can be found here.
After looking around, it seems vscode-database does not host an actual database, but rather, gives you a way to test your queries in the editor. The information page for vscode-database doesn't make that very clear, though. So for MySQL functionality, you may want to look at running MySQL in a "portable mode". Instructions for this can be found here.
After getting that setup, you would just connect to it in php like any other MySQL database.