What were the goals of the Soviet Union and the United States after World War II?

The Soviet Union's primary strategic aim was to prevent an invasion of its vast territories. To that end, it established political and economic control of Eastern Europe, putting in place a series of puppet regimes to do its bidding. The USSR had been invaded by the Nazis during World War II, and the Soviet leadership was determined that there would not be a repeat performance. The Kremlin was concerned that the United States and her allies, whatever undertakings they'd made at Yalta, would try and reduce the Soviet sphere of influence, surrounding its vast Western borders with hostile capitalist states.
For its part, the United States wanted to contain the spread of Communism. A direct confrontation with the Soviets was out of the question given that both superpowers now had nuclear capabilities. So the United States adopted the strategy of containment as an alternative. In practical terms, this involved strengthening the institutions of liberal democracy in those European countries, such as France and Italy, deemed most vulnerable to Communist subversion. By giving such countries massive economic assistance under the Marshall Plan, it was believed that the conditions which traditionally bred Communism—economic chaos and political instability—could be significantly alleviated, thus making it harder for the Soviets to gain a foothold in what was to the Americans such a strategically important part of the world.