This is quite a broad question when one considers that the time period is from 1787 to 1964. One can argue that the civil rights movement continues today as minority groups and women seek equal pay and representation in popular culture.
American politics has been largely dominated with domestic issues during this time period, the exceptions being during times of global instability, such as WWI, WWII, and the Cold War. Tariff law and the role of the federal government have been themes in the national rhetoric, as various politicians have debated what role the federal government should take in the life of the individual citizen. Attitudes have shifted on this—Democrats under the leadership of Andrew Jackson did not back high tariffs or national infrastructure projects, and Democrats under the leadership of Roosevelt promoted large public works and national pension systems.
National expansion and foreign policy have also been key issues. Some presidents, such as James K. Polk and William McKinley, were able to ride the wave of expansion to the White House. Some presidents, such as James Madison, were criticized heartily for their role in wars—Madison was president during the War of 1812, which, while a US victory, did not go well for the US Army. The American people usually elect leaders based on their domestic platforms; however, in times of stress, the people look to strong leadership. Eisenhower and Kennedy were elected at the height of the Cold War for promising to be tough on Communist expansion.
Civil rights have been key in American politics as well, though no one candidate during this time period was elected based primarily on his ability to pass civil rights legislation. The issue of freeing the slaves elevated Lincoln's actions during the Civil War to a holy cause; however, at the time of the Emancipation Proclamation, many in both the military and civilian sectors did not see civil rights as a cause worth defending. The first president to garner the votes of women was Warren Harding. His campaign did not directly cater to women; on the contrary, Harding was a notorious womanizer. Lyndon Johnson was a Southern Democrat placed on Kennedy's ticket in order to ensure Southern support. Johnson was able to use Kennedy's tragic death to pass civil rights legislation, even though it ultimately divided the Democratic party at the time.
There have been many key issues in American politics during this time period of approximately 180 years. Most of the issues have been domestic, though international stresses have led to foreign policy rising to the forefront from time to time. Civil rights and questions of federal authority still remain viable issues in American politics today.