When I'm opening a new file (e.g. with Ctrl+N), the default Syntax in Sublime Text 3 is just Plain Text. I would like to change that into e.g. Markdown. Is that possible?
It is possible with the ApplySyntax Plugin. Once it is installed (e.g. via PackageControl), you can set the ApplySyntax User Settings like that:
depending on your installed Markdown syntax.