Could people with EU pports travel anywhere like with the US pport?

I have travelled to many places but have never had to apply for a visa before hand, it is stamped on my US pport when I arrive in the country. Is it the same for countries that are in the EU? Could they travel to say, America, just by purchacing a ticket and getting their pport stamped at customs? What about the newer EU countries like Bulgaria? Do the same rules apply to all? AGAIN THIS is just for travel not for working!