Red Square - Krasnaya Ploshchad - was called that hundreds of years before communism, and probably means "beautiful" or "shining square", in spite of the red brick walls of the Kremlin citadel which have been there since an Italian architect designed them 500 years ago.
Krasny - "red" - and krasivy - "beautiful" - in modern Russian share a common old Slavonic root. In ancient Russian folklore the heroine was always krasna devitsa - which, scholars insist, means "beautiful girl" and not "scarlet woman". The same word krasno in Serbian (or in Bosnian or Croat) means "shining" or, by transfer, "wonderful". And krasny survives in its original meaning in both Serbian and modern Russian in prekrasny - which also means "wonderful". "Red" as a colour associated with the political left wing goes back to the French Revolution. It appears in that context long before the Russian Revolution in the literature of Turgenev, according to Ig Avsej of the University of Westminster's Russian department who has published translations of Dostoevsky's Brothers Karamazov and Village of Stepanchikovo.
Other colours also have symbolic meanings, though it is hard to pin down where they came from. The Tartar-Mongols, who ruled Russia for more than two centuries, called the newly independent Russian Tsar in the late 15th century the "White Khan" - because, in the Mongol world view, white was the colour of the West. In more recent times, "white" became the colour of the political right. The "White Guard" - the counter-revolutionaries after 1917, for example. And the extreme right, or anyone associated with the disintegration of the Russian Empire, may be called "black" - the "black hundreds" of 1905-1907, for example.
Even more recently, other colours have acquired symbolic meanings. In the Russian language, "dark blue" - siniy - and "light blue" - goluboy, related to golub, a dove, are different colours. Goluboy is slang for "gay". Popular legend is that this derives from the shoulder straps in light or bright blue worn by the secret police. But, as any Russian linguist will tell you, Russian etymology is an inexact science.