Know the Entry Requirements for Individual Countries
A U.S. Passport is all you need for virtually all European travel, but it is a good idea to plan ahead and know for sure. Check with the State Department to be safe. Go to www.state.gov to get started. It is an excellent resource for travel information for U.S. citizens.
Leave the Traveller's Checks at Home
The days of the traveler's check are pretty much gone. Take a credit card, or purchase a pre-paid cash card you can use at ATMs when you get there. Your local AAA office is a good resource.
Plan Ahead For Any Medical Needs
Plan on taking any medications you take normally plus a little extra in case of travel delays. To avoid any complications at customs or airport security, keep them in the original containers with the prescription printed on the container.
Do not spend your time hauling lots of heavy bags around when you could be enjoying yourself. Plan your packing carefully, and check the projected weather for your travel destinations so that you do not end up packing extra clothes that you do not need. For day trips and excursions, bring a lightweight daypack as part of your luggage. This enables you to pack it with some of your clothes and gear for longer hauls or use it as a daypack when you are exploring.
Part of the joy of travel is learning the rhythms of the place and culture you are visiting. For example, you may plan on visiting every archeological site in Greece but find out you are having more fun hanging out in the cafe downtown, soaking in the local vibe and people-watching. Go with the flow and enjoy yourself!
Learn the Language (at least some!)
Many Europeans speak at least some English, but the further off the beaten track you go, the more you will need to know a little of the local tongue. Even if you cannot get all your ideas across, locals will appreciate that you are making the effort. A pocket-sized phrase book is an excellent tool to carry.