It absolutelly is as simple a travelling there and finding a place to stay.
I’ve moved, lived and worked in 3 foreign EU countries just like that.
Those rules about identity documents and registration apply also to the locals: some countries want people to be registered with the city hall of were they live and (supposedly, though in 2 decades I was never asked for it) carry identity papers (though if you have an identity card from your home country that’s valid all over the EU), others couldn’t care less.
You don’t need any kind of visa or even have a job: as long as you can support yourself (i.e. aren’t there to leech of social security) it’s all fine.