what actions did they take in pursuit of these goals that caused Britain, France, and then the United States to declare war?

Germany encroached on places like Austria, Sudetenland, and the rest of Czechoslovakia. Germany invaded Poland forcing the allies (France and Britain) to declare war because they didn’t want Germany to keep expanding while they sat idly by. Japan was in a stalemate with China and wanted to expand, so Japan signed the Tripartite pact (the tripartite pact was a pact between Italy, Germany, and Japan), invaded British Malaysia, and bombed Pearl Harbor. This made Great Britain, France, and the U.S declare war on Japan. Germany then decided to declare war in the U.S, even though they didn’t have to.