Italian working in Switzerland here, I've also worked in France and Germany before, they're all very doable. I do not understand what OP is asking for, an obvious first step for a student would be to take a semester or more abroad, or finding an internship (I've done both, several times, and I suggest it wholeheartedly). OP, having a European passport gives you access to so many possibilities compared to millions of others that all you have to do is find someplace to go and leave.

No, Switzerland should work fine. It may be a bit more cumbersome than a real EU country, but as far as I know, it is still not that hard to move there (if you already have a job). Nothing compared to the US.

Switzerland is now part of the Schengen Area, which allows free circulation of people