There are no requirements for a COVID test for domestic travel within the United States. Internationally, 65 countries have lifted all entry restrictions for travelers arriving from the United States. Another 64 countries allow entry without a test provided you show proof of vaccination. When returning to the United States, vaccinated passengers are no longer required to take a COVID test. (More on that in the answer to Question 5.)
The U.S. Department of State offers country-specific travel information here.
Some countries that have lifted all restrictions for arrivals from the U.S. include: Argentina, Aruba, Austria, Belgium, Costa Rica, Croatia, Cuba, Curaçao, Cyprus, Czech Republic, Denmark, El Salvador, Germany, Greece, Greenland, Guam, Hungary, Iceland, Ireland, Israel, Italy, Jamaica, Mexico, Norway, Poland, Puerto Rico, Romania, Saudi Arabia, Sweden, Switzerland, U.S. Virgin Islands, United Kingdom and Vatican City.
Countries no longer requiring a test for vaccinated arrivals from the U.S. include: Australia, Brazil, Canada, Dominican Republic, Egypt, Finland, France, Luxembourg, Malaysia, Monaco, Netherlands, Nigeria, Panama, Paraguay, Peru, Philippines, Saint Martin, Singapore, South Africa, Spain, Sri Lanka, Thailand, Turks and Caicos and United Arab Emirates.