The goals of the United States and Russia in the Cold War were the same: increase their own power gaining political, economic and military allies.
But both countries were one on each side of the spectrum. Russia had a sphere of influence in Eastern Europe and defended communism as an economic system while the US defended capitalism and liberalism as an economic system.
The United States main goal once the Cold War was installed was to isolate the Soviet Union as much as they could, using all artifacts they could. The US wanted to prevent the SU from spreading communism, this way the US encouraged democracy and open markets around the world.
On the other side, the Soviet Union wanted to spread their ideas of centrally planned economy and communism, the repulse for capitalism.
This led also to the space run, with both US and SU “fighting” to show and gain power in the space.
