For the United States:
* The U.S. (formal)
* The USA (informal)
* United States of America (formal)
* The States (informal)
* The Land of the Free (patriotic)
* The American Republic (political)
For the continents:
* North America (includes Canada, Mexico, and others)
* South America (includes Argentina, Brazil, and others)
For a broader sense of the Americas:
* The Western Hemisphere (includes all countries in North and South America)
* The Americas (a general term for both continents)
Informal and poetic:
* The New World (historical, referring to the Americas after European exploration)
* The Land of Opportunity (suggests a place of potential and success)
* The Great Nation (expressive of national pride)
The best synonym for "America" will depend on the specific context and the message you want to convey.