I have travelled extensively in EU, SA, and APAC.
First, an ATM card will generally offer the best exchange rate, BUT you need to call your bank first to verify any fees they charge for foreign withdrawals. Sometimes the added fees aggregate to a higher exchange rate than you realize.
Second, ask your bank about fees and interest rate for a cash advance on your credit card. Sometimes this may be a better aggregate than an ATM, if you pay off the cash exchange from your checking account the next day!
Third (long term) ask your bank if they offer (or find a bank that offers) a Travel card that will not charge any foreign exchange (FX) fees. These can often be the best all round option as you can use them in store, restaurants, etc. and avoid most foreign fees.
Fourth, cash: The worst cash exchange rates are found at the most convenient locations: Airports, Hotels, Tourist venues. The best rates can be found with some research, typically in small "boutique" booths in malls, or on high-foot traffic local roads. Look up the current FX rate for your currency on a Xe Currency Converter (use the link) and know the "base" rate at the time you want to do the exchange. The service will need to make some money, so expect them to pad the offered rate by 20-60 BPS (=Basis Points. A Basis point is .01 of a percentage point. So 100 BPS is 1%). Businesses that offer currency exchange in addition to their main business, like selling phones and electronics, will generally add fewer BPS as they do this to attract business, not as a profit driver.
About using cash: Exchanges will want crisp, completely unmarked $100 bills. They may reject bills that are very old, torn, have pen marks, rubber stamps (like "Where's George"), etc., as potential counterfeit. They will usually also accept $20 bills in the same "new" condition but may charge a higher rate (5 to 10 more BPS) as these bills cost them more to handle and have additional risk of fakes.
Finally, make sure you spend all of your EU cash in the last few days of your trip, even if you have a card available. Don't forget that you will want to exchange your Euros for USD on the way home, and you will likely end up doing that at the airport before you board. That means you will end up double-paying: once to get your Euros, and again to get your USD back. The less cash you have when you leave, the less double fees you will end up paying.
Life is good
The Cat