While most places in the U.S. do take Amex, it's worth keeping in mind that some major retailers, like Costco, don't accept it. At Costco, your best bet is the $0 annual-fee Costco Anywhere Visa® Card by Citi*, which doubles as a membership card with cash-back benefits.
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Here are some countries that don't accept American Express at all:
If your goal is to get or maintain a good credit score, two to three credit card accounts, in addition to other types of credit, are generally recommended. This combination may help you improve your credit mix. Lenders and creditors like to see a wide variety of credit types on your credit report.
First and foremost, you want to ensure that the credit card you take to Europe is widely accepted. Generally, the Visa and Mastercard networks have wider acceptance there than Discover and American Express. While you can certainly bring your Amex and Discover cards, it's best to pack a backup.