If you plan to travel to Europe for a few months, make sure you are aware of the Schengen rules, because these rules dictate how long you can stay in Europe. Some people plan extensive trips through Europe without realizing that they are not allowed to stay for months and months on end.
Most, but not all, countries in Europe are part of the Schengen Agreement. In total, the agreement was signed by 25 countries. The most notable exception is the United Kingdom, which is not a part of the Schengen Agreement.
Countries included in the Schengen Agreement are: Austria, Belgium, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, and Switzerland.
Basically, as a US visitor you are allowed to travel freely between any of these 25 Schengen countries for a period of up to 90 days (out of every 180 days).
That means for a 6 month period, you can spend 3 months in the Schengen area.
Here's an example of how it works: If you want to travel to Italy, France, and Spain all within a 90 day period, that's fine. However, if you want to stay in Italy for 2 months, France for months, and Spain for 2 months, you are technically not allowed (unless you acquire a long-term visa, which you would need to do beforehand).
Understanding the Schengen Agreement is pretty straightforward. Just make sure you plan accordingly!