Well I moved here for school so the process was pretty straightforward (student visa). There's a lot of paper and procedures to keep track of but it's not insurmountable.
As far as culture shock goes I spoke the language well enough to get by just from school, but there were definitely some issues. (I still run into a few things now and then where my entire worldview is diametrically opposed to most of the people around me, but it's much rarer than it used to be, haha.)
As far as specific stories go...I do remember thinking American college kids were incredibly coddled compared to university students in Europe. (The fact that it's okay to refer to them as "kids" should have been a hint.) Where I grew up, when you go to university, you move to a new city and get a studio or and apartment (perhaps with a roommate) and your life is basically not that different from a young professional in terms of schedule and structure. You make your own food, go to lectures of course but mostly study independently and treat going to the library as your "job". There will be some activities in the beginning of the year to get to know the other students, and you might join a student organization if you want, but if you want to sing in a choir or go to a gym or join a political party you're usually surrounded by all sorts of people, not just students. In the US, however, my college peers were living in a dorm with an assigned roommate, eating in the dining halls, playing sports with other students, doing drama with other students, and discussing politics with other students. Some of them seemed to go months without interacting with anyone who isn't a college student or lecturer. It blew my friends' minds that I lived off campus in a privately rented apartment and cooked my own breakfast every morning. There is this idea that college is a separate life stage where students live in a small bubble and it doesn't have anything to do with "the real world". Some of my friends called their parents every day and got lectured if they skipped class. Their parents weighed in on their class schedule and expected to hear the results of every test.
(Just as a disclaimer, I understand why these differences exist now. It was just weird for me to adjust in the beginning.)