im going to try and clarify a couple misconceptions in this thread. correct me if i am wrong.
A US passport will allow you to travel to many countries without a visa. even countries like Spain you can travel to without a visa. if you are planning on studying or staying in a country for an extended period of time, you will need a visa (i know this is a fact for Spain and Denmark). a visa may be enough for working overseas, but some countries require a work permit (i know this for a fact of Denmark) which you will need when applying for a job. on my US passport, i traveled to the England, the Netherlands, Switzerland, Spain, Greece, Germany, Austria, France, Italy, Denmark, Norway, and Sweden without any problem.
A us passport will grant you all of your US rights at any US Embassy also. If you were to find yourself in any kind of legal trouble, find your way to the nearest Embassy and the will help you out. They are highly sought after documents so as stated before, be sure to keep it with you and secure at all times. I did lose mine in the Zurich airport when flying back to Denmark, but since i had already gone through security and had a Danish student ID, they did still let me fly after some pleading.
read the government sites that have been linked, but also check out foreign travel sites as they will tell you exactly what you need.