EU is not a single country, as the US. Each country has its own rules and regulations.
Some countries, as mentioned, offer a digital nomad visa (Croatia is one of them) others do not.
There's also something called the Shengen area, which allows for free movement between countries. Not all EU counties participate (yet) and even some non-EU countries are part of it. This allows you to have a citizenship in one country and live / work in another.
As for the language, your mileage can vary. Most countries will require at least a basic language knowleget to pass the citizenship test. For day-to-day communication, work and study, you can find options where English would be sufficient. Lots of universities offer English-language programs and lost of companies are international with English being the primary language. Most Europeans speak English quite well.
As others have mentioned, a student visa for one of these would be your best bet. Explore which country would make most sense based on your needs.
If student visa is not an option, then getting a job (in the target country) first would be a way in. Get a working visa and start applying for citizenship.
You should also understand that literary everybody ion Europe (not just EU) has an ID and is registered as a citizen. It's therefore practically impossible to be undocumented and still get salary, open a bank account or visit a doctor.
TL;DR: Explore each country individually. "Moving to EU" makes as much sense as saying "Moving to Asia".