As far as I remember, the Visual Studio Installer project can be configured to include the installation of other components like the runtime library as part of the install. It won't embed the whole thing but rather download and install the latest from Microsoft's servers during the install process.

Look it up. If not, there's Inno setup. It is a powerful, script driven system that definitely handles pre-requisite installs.

I have never done this before, I currently just create a zip containing all the files, but that's not the way it should work

If you do it this way you will not be registered correctly for install/uninstall and app pool. On windows you can do all things by a custom install application, that means - check for dot net ramework if so, run vs redist, invok setup.exe with parameter to install your app correctly, all done in your custom interface. Also in registry create desktop icons, run on start-up and such.

You can do that, or you can write your own. It's all resources, registry settings, temp files, running other installers and public Windows API.

If you want to do the installer quickly, use Inno or NSIS. In case you are curious how it works on the inside, or you want to heavily customize the UI, you might have to roll your own.