I use my Amex quite frequently but also have a Mastercard and Visa just in case. But cash is still king in many countries, such as Germany. And as someone pointed out, chips are common in credit cards here; hence your American cards often won't work in machines at train/metro stations, parking machines, etc. As someone pointed out, don't expect to pay in American money, and don't tip in it either. You wouldn't like it if I gave you a few Euro coins, right?
Are you interested in 'Europe', or specific countries? 'Europe' is rather vague and too broad to give information. If you narrow it down we can give you country specific advice. From what I gather - London, Paris, and ??? If you are including Rome, for instance, I recently did a review of the various tours of the city available and can link it for you.
I am in Paris every few months; don't know what you want to know. I strongly urge people not to follow Rick Steves advice as I have some very strong feelings about him. If it is your taste, fine, but you won't see as much of real people and places - but you will find lots of Americans around you (I passed a restaurant in Rome a few weeks ago with a sign 'featured in Rick Steves' and it was packed with Americans, with a menu board in English. Not very 'local flavour')
For Paris don't get sucked into the overpriced Paris Visite card; buy carnets of metro tickets or a Carte Orange for weekly travel depending on your arrival date. Check the
www.ratp.fr site for travel info, but note that if you turn it into English language it won't give you the cheaper options but only the overpriced tourist options.
If you are buying train tickets buy them at the local sites ie
www.sncf.fr etc. The North American sites which sell European train tickets have a high markup and you will save a lot of if you buy local. Look for local specials; I can often get train tickets from Germany to Paris for 29 Euros, for instance; or I can find First Class tickets for less than Second Class if I am flexible on date/time. Flying may be cheaper than taking the train in some cases; you really need to do your homework. But note that there are dozens of LCCs in Europe and if you add in the cost of getting to out of the way airports, fees for checked luggage, and fees for boarding, low cost carriers may not be cheaper than legacy carriers. If you then factor in travel time, flying may be a better value than taking the train.
Americans often find our hotel rooms small; many do not have air conditioning and most do not have washcloths, unless they cater to Americans.
Eat like the locals; that means find the local bakeries or local chains or even food from the grocery store. In Paris the bakery chain Paul is excellent for cheap meals; Hippo is a good local restaurant chain. There are fixed price menus in most restaurants which are a good value. Ask your hotel or people in shops etc for a recommendation. Paris doesn't have to be expensive, contrary to popular belief.
Buy souvenirs at the grocery store. Local chocolates, cookies, drinks, etc make good gifts at a reasonable price. Look for local hypermarkets such as REAL in Germany or Auchan in France, or local discount stores such as Monoprix. They usually sell better products at a reasonable price when compared to places aimed at tourists.
Clothing and shoes are expensive here; I can buy Longchamp and GEOX for instance, two of my favourite brands, for cheaper in America than in France or Italy or Germany. If you are interested in buying clothes or shoes, then do some research but you may be shocked at the prices. Sales are usually held twice a year in January and July in Paris; this is a good time to stock up on end of season items.
Don't feel that you have to 'do' sights. I see tourists trailing around, looking miserable, because someone told them that they had to 'do' Versailles, or 'do' the Louvre, and they have actually no interest in art, or history. (And those sights can take many hours or even days to see even part of what they have to offer) Follow your interests and include them in your plans. If you don't like going to museums, and have no interest, then find something more to your taste. Buy local guides in the cities, such as 'Time Out' or similar. For guidebooks, find one of your home town, read the description, and if you like what they say and it is accurate, that is an indication that the series is going to be useful. I often find books which are inaccurate, or even offensive, about where I live, so I won't buy books in that series.
Avoid being scammed; American tourists are a target in big cities because they are visible and sometimes too friendly. If someone asks you to help find their diamond ring that they dropped, or a woman runs after you crying 'speak English?' chances are they have identified you as a possibly easy target. Don't feel bad about saying 'NO', or just walking away. I never get bothered by these types because I don't stand out and don't make myself available to them.
Dress appropriately. All summer I saw tourists turned away at cathedrals (even at the Vatican!) for wearing shorts and bare shoulders. Short shorts, tank tops, flip flops are not worn in cities by most locals and may be considered inappopriate or even disrespectful.
Learn some basics of the language. 'Hello, goodbye, thank you, excuse me, do you speak English?' Don't launch into English and expect us to understand. Even if we do speak English, many of us struggle to understand some American accents. It may sound like a stereotype, but again I saw this all summer long and actually had people shout at me in English without even addressing me in German. If I walked up to you in your hometown and started babbling at you in German, you may not be overly friendly or helpful either. People of an older generation, or in smaller towns, don't always speak English. Even some Germans I know don't speak very strong English, so be aware of that.
And remember that most of us do
understand English, so don't talk about us loudly and criticise our clothing, or habits, if we are sitting right next to you on a train or in a restaurant. Yes, I had that happen to me many times this summer too... apparently I dress 'weird' as do many of my countrypeople.
Again, if you have more details as to what specifically you wish to know, it would help to narrow it down. I hope that I didn't offend anyone in my comments; I am always happy to help people plan their trips, but this summer in particular seems to have been a bit of a challenge and I will admit that I am a bit cranky from some of my interactions...
I pay for my gas in Germany with Amex all the time. I pay for a LOT of things with my Amex here. I even have one for work expenses and one for personal expenses because it is so useful. In fact I just was in Germany, Austria, Belgium, France, Italy, Switzerland, Monaco in the past few weeks and used my Amex for several thousands of Euros in purchases. I even use Amex at small bakeries etc. 
) here as well, but you won't find them anywhere near the tourist areas. You have to go outside the downtown city location. It would take you about 1-1/2 hours to get to the one here from Downtown. You would have to stick with the mom and pop type places if you have no car and you are not adventurous.
Be specific when asking for water. Italians drink fizzy water. Just say "Naturale or no gas". Also, most sodas are pretty close to room temp and wine is ALWAYS room temp. The house wine is usually cheap and very good. If buying wine from a store, don't pay more than 10Euro a bottle. Good wine does not have to be expensive as assumed in America.
