What is the most economical way to exchange US dollars for euros?

Answer 1

If you travel to Europe, make sure your debit or bank card is good for ATM use abroad. Get one that does not charge for withdrawals.
You get money in Euro but the bank statement will mention the amount of dollars you used. (You get the best rate that way.)
If you need to pay bills in Europe, like a hotel bill, you can use a credit card, when that transaction is booked depends on your card. The cost is mostly small, but if you use a credit card to get cash, you often pay a lot on surcharges.

If you do not go to Europe, or if you do only go for a short time, you might want to change at a bank office or an currency exchange office.
Mostly they charge a cover charge, one amount of money however much or little you change.
Some have a no charge policy, they are better if you want to change small amounts. They pay a lower exchange rate but for small amounts it is worth it.

The least economical way is buying traveler cheques, you pay to get them, and you sometimes pay to get your money back. And you still loose on the exchange rate, at least as much as when you change at a bank office.
Besides, most places in Europe do not like traveler cheques, Hotels will charge to handle them, and there are few bank offices that still deal in them.