I live in Europe but I think I may as well provide some informations.
My recent travel history
1997 Rome, Italy
1998 Paris, France
1999 Slovakia (bicycle tour)
2000 Barcelona, Spain
2002 Cote d'Azur
2003 Vienna, Austria
2003 Denmark (mainly Copenhagen, Aalborg, Odense)
2004 Ljubljana, Slovenia
2006 London and Riga, Latvia
several times in northern Italy and in Germany
several times in western Austria for skiing
Because of its dimensions and me being a leader in countless scout camps I also know many if not most of Switzerland's corners.
Guidebooks. A good preparation includes a guidebook, in my opinion. There are huge and thin ones, better and below-average ones... The size of a guidebook is a matter of taste. I seldom had bad experiences with them (p.e. when I was in Riga, my guidebook suggested to visit the west of the city for its great architecture, but the center turned out to be much prettier) - but they help to remember small things during the preparation of a trip that could be forgotten otherwise. If your guidebook has no or only a small city map in it, you can get a bigger one for free or a small price in the tourist office, which mostly is in or near the central station.
Language. Don't miss the countries where English isn't national language, especially if you've already been in Europe. English is spoken to different degrees by the people. However, where it is crucial (hotels, hostels, campgrounds, informations for tourists), people will almost always speak English. It's not wrong to know some basic words in the national language either (like yes, no, thanks, water) either, another thing guidebooks are good for.
Outfit and behavior. What a title

. Well I have nothing against those that fit the stereotype of the always sunglasses and camera-wearing tourist and so does the overwhelming majority of Europeans (I may come close to this stereotype myself, sometimes). But your outfit should be appropriate to the climate, so inform yourself. Groups of Asian tourists with short pants and sneakers get some strange looks on the top of the Jungfraujoch-- 25°F, from time to time. Okay now for the behavior. No, you don't have to pretend to be from Canada and you don't have to discuss American foreign policy every five minutes (but of course you are free to do so). Don't be Homer Simpson-ish and point at persons. I mean: act naturally and normal. For example you should not try to swim in the Fontana di Trevi in Rome (you are not Sophia Loren, after all). If the police sees you, they will give you a fine.
Cultural differences: the small and the tiny. The differences are small but they exist. It is not necessary to memorise everything but be prepared to encounter them every now and then. Among the more important things are the tipping habits that vary through Europe. And because I'm always a bit representing my country abroad, I try to get them right. Also, I've been told that Europeans tend to sit a bit closer to each other than Americans in restaurants and bars and that other than Americans they often slightly have physical contact with each other when entering or exiting elevators and trams. I haven't tested this myself though. Just keep in mind that the size of the space between you and your neighbor or your vis-à-vis' table manners aren't directed against you, but common where you are
Travelling: train, plane and pain... Personally I'm a strong advocate of travelling by train. It is comfortable, there's more space than in a plane or car, and long-distance trains usually have a dining car. And looking at a landscape is better than just seeing clouds or the pavement of an autobahn, isn't it.

The Eurail ticket has been mentioned, it is a good offer and I've used its equivalent for resident Europeans for my Denmark trip. There are some cheap airlines for continental flights. For the first time I've used them three times during my London-Riga vacation (
RyanAir and the already mentioned easyjet) and it has cost me $80-100 each flight, all taxes included. But I wasn't very impressed. Our airports in London and Frankfurt were in the boonies, often 1-2 hours from the city center (which is somewhat dissappointing because in Zurich it is 15 minutes by train and we missed one flight because the bus had one hour longer than normally for its way to the airport). So prepare to spend some additional money for the busses and calculate with a big time reserve for a worst case scenario if you chose to fly with cheap airlines. Of course it's possible to rent a car for your trip but I lack experience there. Parking lots and gas cost more than in the US, however.
Theft, no-go zones. Regrettably there are persons who try to rip off tourists, you'll encounter them often or rarely depending where you go and which places you visit there. Normal pickpockets, teamworking thefts that use distracting maneuvres and others. They may attempt to slice your backpack open with a razor blade or to shortchange tourists that aren't familiar with the local currency, confusing them with many coins before they disappear. Meh, don't let them ruin your stay. Just pay attention to your wallet and valuables and you'll be fine. Use hotel safes if you need to have many valuables with you. Regarding no-go zones, I don't remember any spontaneously. But it doesn't take Spiderman senses to see which are the dubious quarters of a city if there are any.
Festivals and events. Does it make sense to plan to visit a city because there's some sort of festival? Normally no, in my opinion. However, after you've decided which places you'd like to visit, it is wise to look up if an event that makes you curious happens to be hold there, you could shift the timeframe dedicated to the city. On the other hand, if you prefer to experience the everyday life in a city and don't check for events, you may stumble across a million visitors of Zurich's Streetparade or a horde of soccer fans that is crazy for the world cup final in Berlin.
Museums. Boring and only for old people? I say no, or not necessarily. If you have a bit more time than just two days, consider the visit of a museum. Not everybody is into art, that's true. But history museums can provide you with new perspectives. For example when I was in Riga, the National and War Museum was very interesting (although somewhat depressing due to Latvia's changeful history). Museums are a good way to learn about history on site - And if you don't like the museum you can always leave right after you enter and maybe visit a museum that fits your special interests (technology museums?). Also, museums are mostly cheap or even free.
Electricity. Most power supplies for cell phones and laptops accept both 110 and 220 V input nowadays, but you may want to check. There are differently shaped power sockets in Europe, so better look up if you need an adapter for where you go.
Food. American-style fast food is common in many places but no travel is complete without some culinary experiences. In some countries like Italy, national specialities are offered everywhere. In others, the national kitchen is more hidden or you may have to visit a bakery or a pastry shop instead of a restaurant to find them. Yet another argument for the guidebook.
Cost of living. As a rule of thumb, life is cheaper or the same as in the US east of the former Iron Curtain and the same or more expensive in the west. However depending on membership in the Euro-zone and the national performance of the economy, the price level changes over time. Cities are generally dearer than the countryside. However, depending on your lifestyle, it is without a doubt possible to have cheap holidays in the west or to burn a huge pile of money in the east.
Drinking water, tap water. In some countries wells don't have a warning if they don't provide drinking water (my city on the other hand is proud that it has 1200 wells that provide free drinking water). The quality of tap water depends on the region you visit. It may be just fine. Or it may taste chlorinated. In few places visitors are even advised to not drink it. If you're not content with the tap water, buy a bottle of mineral water at the local superstore. It is cheap and always okay.
Less! Warning! Signs! It may be a normal cultural difference. However I want to clearly point out that although Europe's catching up, there are way less signs here. To the average European it is understandable that objects appear smaller in mirrors, or that you shouldn't climb a tower if you have a heart condition. Mmkay?
City visits and beyond. City trips are easy to organise, leave you a big flexibility, and often are cheap. But I thought I could name some other possibilities to travel Europe. Winter resorts and skiing or beach vacations are one way to do it. You're still flexible, but it's not that cheap. And then there are possibilities ranging from unconventional to exotic. Bicycle and walking-tours of all sizes, camping and so on. These alternatives come in all price levels but usually require some serious preparation. Especially if you plan to visit sparsely populated areas or mountainous terrain. An unplanned rescue by helicopter more than once has burdened a vacation budget in the Alps...