It's been hinted at in dan_sim's post, but if you have a "real project" in mind just go do it. Learn what you're missing when you need to. You probably know more than you think.
But there is nothing to show me if I correctly handle my prod-server updates (Now I upload, move, restart)
My testing is very basic.
And there is a general feeling of lack of infrastructure.
I would love to learn how to do a few-man-year project (from infrastructure's point of view)
Have you got any hobbies or interests? Why not create a website dedicated to your hobby/interest? That way it'll be something that you'll be able to run over the course of a few years.
Start with a small website, and slowly build it up. Add more features, then tweak them. Find out why it's loading slowly. Read a few books on website that scale too.