There are four international Disney resorts falling under four jurisdictions with their own travel restrictions. While the rules for visiting these Disney resorts as a U.S. passport holder are mostly simple, it's not quite as easy as visiting the parks within the United States. (Heck, I don't need to worry about getting from the airport to Walt Disney World when I go to Orlando!)
Here is a quick overview of the visa requirements for visiting each of the four international Disney resorts. Because you don't want to mess up visa requirements, you should definitely confirm these rules before booking.
Shanghai Disney Resort
Shanghai is a city in China. U.S. passport holders seeking to visit Shanghai Disney Resort will (generally) need to apply in advance for a tourist visa (L Visa). To obtain a tourist visa, you'll need to provide your local consulate with documentation of your hotel stay and your air travel itinerary (including flights from and back to the United States).
The fee is $140 (per person), and the processing time is four business days. If you are just passing through Shanghai, you might consider looking into their 144-hour transit visa, but this is only an option if you visit a third country (not your origin or China) after Shanghai.
Finally, it is probably worth specifying in your application that you would like a 10-year, multiple entry visa. This is a relatively new offering from China, where for the same fee ($140) you can get a visa that is good for ten years.
Tokyo Disney Resort
Tokyo is the capital of Japan. U.S. passport holders can travel in Japan for 90 days without a visa, but they should be prepared to show proof of a return or onward ticket.
Hong Kong Disneyland Resort
Hong Kong is technically a "Special Administrative Region" of China, but it has its own set of travel/visa rules. This is nice for U.S. travelers because it is easier to travel to Hong Kong than to China. U.S. passport holders can travel in Hong Kong for 90 days without a visa, but they should be prepared to show proof of a return or onward ticket.
Generally, U.S. passport holders can travel in France for 90 days without a visa. However, as France is part of the Schengen Zone of Europe, you can only spend a maximum of 90 cumulative days in the Schengen Zone in any 180-day period. This means if you spent 90 days touring Europe in June, July, and August, you cannot visit Disneyland Paris in October. You would have to wait until early December.