You can't "just" become a citizen, in most countries. There's usually a process, requiring clearing hurdles like being a resident for a specific length of time, working, etc. Differs by country. Most, because some countries sell citizenships to those rich enough (EDIT: Malta, which is in the EU, offers full citizenship for around $1million, lol).

As for travel, you (US citizen) can reside for up to three months visa-free within the Schengen area (the borderless zone within EU and close partners). Outside of it, it'll differ country by country. If you need more details about visas and traveling in Europe, there are plenty of official online resources, including advisories on your government websites. I suggest looking at those and NOT relying on "some guy who went to Europe 5 years ago told me", because these things have a tendency to change. Stay up to date or risk nasty surprises.