The biggest thing to keep in mind if you're an American selecting a European club to support is that they are not sports franchises like exist in the US, they are "clubs" and they each have developed uniquely over time. This means that in many cases supporting a club is also making a political statement, or outwardly affiliating yourself with specific ethnic or religious groups. Supporting a club is something that says a lot about you, far more than just "I like 3 players, so I cheer for the red shirt with the black stripes". There's nothing in American sports (well, college athletics to a far lesser extent) that resembles this.
So watch a lot of matches, and research the clubs very carefully. It's not something you have to jump into right away, take your time and make sure you're identifying yourself with a group that largely matches your views and beliefs - the team on the pitch is just a public extension of the group of supporters. That's who you are really joining.
Are you eastern Orthodox? Is that the most important identifying characteristic of your life, and you are generally relatively socialist in your politics? Then Olympiakos (Greece), Spartak Moscow (Russia) or Red Star Belgrade (Serbia) should appeal to you.
Are you related any former KGB or other security forces members? Are you a supporter of strong, autocratic governments? Anyone with Dynamo in their name (Dynamo being the former KGB sporting club) is set for you. Dinamo Zagreb, Dynamo Moscow, Dynamo Kiev, etc
You think Hitler had some good ideas? You wouldn't mind seeing a return of some form of fascism? You think Jews are dirty thieves that are ruining the world? Chelsea, Lazio, and a host of eastern European clubs (Lech Pozan, Zenit St. Petersburg, etc) are right there for you.
Are you Jewish? Do you self identify as one above anything else, or regularly find yourself grouped with them? Tottenham or Ajax are your likely homes.
Are you a blue collar, pro-union guy? Do you like being part of a big group of relatively like minded people from "looked down on" cities who are struggling a bit in the post industrial western world? Say hi to Liverpool!
Are you a punk, a queer, a general anti-establishment type? FC St. Pauli is home to any and all - as long as you aren't right wing.
There's tons more - each club in Europe has it's own history and identity that has accumulated over time. Pick the one that speaks to you.