I understand that Europe also has a much more robust system of passenger trains, which is basically non-existent in the US
Most of Europe does have a pretty good mass transit system. Cost really depends which country we're talking about and even then, how far in advance you buy tickets matters. Major cities in Europe tend to be old and unfriendly to driving (Prague, Munich, Berlin, Nuremburg, Amsterdam) so in city transit and walking are necessary. People in Europe tend to bike more than Americans as well. The Netherlands has huge bicycle only parking garages. Uber is a thing in some places but not ubiquitous.
If you're coming to this side of the world, please do your homework. There are plenty of things that are very familiar to Americans right next to an opportunity to make a total ass of yourself. Think through your travel in and out of Europe ... how you'll get around ... what you'll see besides hockey (if you go to Prague and don't see the sites I will want to beat you with a hockey stick ... it is an absolutely amazing place with great food, drink, historic sites tied to the start of wars, beautiful architecture, museums, etc) ... if you'll stay in a local hotel or stick to the more American style chains ... be prepared to carry cash in some places because you might not find a place that takes your credit card and check with your credit card issuer to see what foreign transaction fees they charge, if any ... understand the money conversion rate and if you can get the 17-21% sales tax back as you leave the EU ... and if you have questions ask the European fans or the Americans living abroad for how to maximize your trip.