I've read about how the US used propaganda in Europe to convince the populations in the european countries that the american way of life was superior. I've also heard about how the soviets used supported the expansion of communism in other countries.
Did these two countries use propaganda on their own populations in order to keep them from getting influenced by the opposition?
In short, yes massively so. Huge amounts of propaganda were churned out by both sides for domestic consumption.
There are so many examples it's hard to know where to start but as an illustrative example, lots of Hollywood sci-fi B-movies in the 1950s were obvious allegories for communist infiltration - such as Invasion of the Body Snatchers.
On the USSR side, the @sovietvisuals account on twitter is full of examples of poster campaigns aimed at the Soviet citizenry - covering everything from agricultural policy to the space race.