At the end of WW II the USSR seized the Baltic states (Estonia, Latvia and Lithuania), which had succeeded in separating themselves from Russia in 1917-19. The USSR also put into power governments they could control easily in all the countries they liberated and kept massive numbers of Soviet troops in them. Disagreements with their Allies led to the Berlin blockade, the Berlin Wall and the Iron Curtain. The Soviets forcibly put down a rebellion in Hungary in 1956(and Czechoslovakia later, in 1968).
The obvious efforts at continuing the World Revolution included bringing political figures from other countries to Russia for education and political training, especially at Patrice Lumumba University in Moscow, and support for revolutions in the Congo, Kenya, Malaysia, Cuba, and many other countries. They also supported the Chinese communists during the final phase of the Chinese Civil Wars, and in helping North Korea. They also supported wars in Viet Nam and neighboring nations all through the late 1940s and '50s, and of course later.
Less obvious were the cultural offensives, including Youth Festivals and limited, tightly controlled trips by ballet troupes and other artistic groups to the West. Of course, there were also continuing espionage activities on the part of the KGB, most ruthless and perhaps the most effective spy agency of modern history. The McCarthy era gave the Soviets plenty of opportunity for disinformation activities, and there were "defectors" who were plants as well as true defectors.
In addition to the above mentioned actions, the Soviets were very active through the COMINTERN, or Communist International, from 1945 - 1960 to try and spread communist rebellion in other countries.
Besides instituting communist governments in Eastern Europe and putting down the Hungarian uprising, they also sponsored Fidel Castro's revolution in Cuba and gave him substantial foreign and military aid. They expanded the presence of the KGB spy network throughout Europe and the United States, and they completely armed and sponsored the North Korean invasion of the South and the subsequent war from 1950 - 1953.
They fomented rebellion in Greece and Turkey in the late 1940s, helped China achieve communist independence, and planted the seeds of communist movements in Africa, Latin America and Asia.