Explain US foreign policy from when the country was founded to the end of the Cold War.

This is an enormously complex question, but we can make an attempt at answering it by dividing a 215-year period (1776 to approximately 1991) into five periods or phases:
1. 1776 to 1815. The first forty years starting with the Declaration of Independence saw the new country in survival mode. Congress knew the only chance to win and sustain independence was to achieve foreign recognition and an alliance with France, Britain's perpetual enemy. Independence and sovereignty became real with the Treaty of Paris in 1783. After the Constitutional settlement in 1787-88 and Washington's taking office as our first president, policy had to be directed to avoiding involvement in the huge disorders that began in Europe in 1789 with the French Revolution. Both Washington and his successor John Adams wished to regard the War of Independence as having been a family quarrel with Britain, and our one-time allies the French became quasi-enemies during Adams's administration. But under Jefferson and Madison, the orientation shifted back against Britain. The resulting War of 1812 settled one thing: the British would never again attempt to interfere with the new country or continue their desultory attempts to provoke the Native Americans to attack the settlers.
2. 1815 to 1898. This is a long period, but it is characterized by a consolidation of US power in the North American continent as a whole and the resulting conflicts with those foreign countries who either interfered with US ambitions or sought to destabilize our neighbors. The Monroe doctrine was a warning to the Europeans to "stay out." The Mexican War was a (successful) attempt to expand US territory and to weaken Mexico as a rival power interfering with the already independent Texas or, potentially, areas already within the US. In 1862, the French took advantage of the US Civil War to take over Mexico, but given the Union victory in 1865, the unprecedented power of the US armed forces, and US opposition to Napoleon III's attempt to re-colonize parts of the American continent, the French could not sustain their venture and were expelled by the Mexicans.
3. 1898 to 1917. This is the first period of tentative outreach beyond the Americas and of US armed involvement in other countries' affairs in those distant places, during the Spanish American War. The US now had the will and the ability to take non-American colonial territory, as was done in the Philippines.
4. 1917 to 1945. The enormous events of this period show US foreign policy having direct military involvement in Europe, first in the last year and a half of World War I, and then in World War II from 1941 to 1945. In the Second World War, the US also expelled the Japanese from those areas of Asia—the Pacific Islands, the European colonial territories, and China--which the Japanese had taken over. The US was now an international power, emerging finally in 1945 as the strongest country in the world—the "leader of the Free World"—as both the European and Asiatic continents had been devastated by years of warfare.
5. 1945 to 1991. US, British, and eventually Western European foreign policy, once the countries on the Continent were back on their feet, was aligned in the Cold War against the Soviet Union from 1945 on, and against Communist China as well from 1949 on. The whole policy was to "contain" communism, and the Cold War became a real war first in Korea and then Vietnam. Ironically the failure of the Vietnam effort coincided with the beginnings of rapprochement with both the Soviets and the Chinese during Richard Nixon's administration in the early 1970s. After the interlude of the dovish Jimmy Carter (1977-81), Ronald Reagan's hawkish stance and deliberate intensification of the arms race were given credit for forcing the Soviets into their policies of glasnost (transparency or openness) and perestroika (reform). The European satellite countries were let go in 1989, and two years later the Soviet Union itself fell apart. The Cold War was over, and Americans celebrated the "end of communism" in spite of the fact that China was a totalitarian state with a billion plus people. New conflicts had already begun to emerge with the Iranian Revolution and with the increasing disorder in the Middle East generally.
The above is merely a sketch but, I hope, a helpful one in explaining the transformation of US policy during the country's first two centuries and beyond.