What software do I need on the Web server to Angular 2 / typescript app to get it working? Or I can just upload the files to the web server?
Do you use angular-cli? (I'd recommend you to use it if you haven't). Because you just can do this command on your terminal
ng build -prod --output-path=path/to/home/of/web/server/
What it does, it builds your app and generate
dist folder, in this case because you specified the output, it directly generate the dist content inside the path.
Hope it helps.