It's difficult to choose just four events from this time frame, as the Cold War intensified and spread during that decade. In addition to the events stated in the above posts, which are all great, we also saw some other very important developments at that time in the ongoing competition between the US and the Soviet Union. Four I would emphasize are:
1) The beginning of the Space Race with the launch of Sputnik and the passage of the National Defense Education Act
2) The CIA-aided overthrow of Jacobo Arbenz Guzman in Guatemala as the US expanded efforts to stop the spread of communism into the Western Hemisphere (though Arbenz was more interested in simple land reforms as opposed to being a full-fledged communist
3) The uprising in Hungary in 1956, ignored by the West because it did not fit with the policy of containment. It was crushed by Soviet tanks.
4) The Cuban Revolution - Castro's rise to power, first supported by the US, until he announced his alliance with the Soviets, marked the first country in the Western Hemisphere to go communist, and gave the USSR a foothold for rebellion in Latin America. This would cause a series of events into the next decade and administration, from assassination attempts to the Bay of Pigs planning, to the future Cuban Missile Crisis.