I'm a US undergraduate in math and physics and I'm looking to ultimately do a PhD in either physics or math. It seems that a lot of the research I connect with the most is happening in Europe. I also like the focus on research rather than coursework in European PhD programs. However, I've found most of the process confusing for an outsider like me. 

I would like to know the ins and outs of how graduate programs work over there. There's different lingo there, for example: Taught vs Research MSc. Which is better ultimately for a PhD program? I also get the feeling that Master's and PhD in Europe mean something totally different than what we mean in the US. Are there funded Master's programs in Europe (in math/physics) as there are in the US? How does funding compare?

For reference, I've been looking at schools in UK, Germany, and Switzerland (although not so much Switzerland due to the high cost of living). 

Thanks!

Edit: I'm now aware of how broad my question is. Nevertheless, I found much of your advice helpful in gaining a general knowledge of how to pursue postgraduate degrees abroad and some additional considerations, so thanks! If/when I have more specific questions about this process I will ask them in a different thread.