How did the US and the USSR compete to spread the ideals of Capitalism vs Communism around the world? What was it like for other countries?

This is an immense topic, and is basically the scope of any work examining world history between 1945 and 1992.

I recently attended the presentation of the latest edition of The World Since 1945 cowritten by P. M. H. Bell and M. Gilbert. That could be a good place to start.