An /r/AskReddit thread is full of claims, and this exchange stood out to me:
Did the Russians really want to "win" the "space race" though? I remember from a previous thread getting a man to the moon being the goal was more an American invention after Russia did almost everything else space based first. I am sure the Russians probably felt they quite soundly won the space race.
Followed by:
The notion that there was a space race was an American invention. Sputnik was not considered a big deal in Russia until Russians realized that Americans were making a huge splash about it.
How accurate are these assertions?
Did the Russians really want to "win" the "space race" though?
Nope - "space race" existed only in western media. "Moon race" popped up in late 80s.
The notion that there was a space race was an American invention.
True. You won't find any notions of "space race" in soviet media - except occasional jabs of "USA scramble to catch up to USSR space program", "gimmicks like putting men on the Moon just for the sake of it" - as opposed to sending USSR moon rovers Lunokhod "for science".
Sputnik was not considered a big deal in Russia until Russians realized that Americans were making a huge splash about it.
Not true - USSR media reported it on front pages immediately and prominently. And kept reporting it.
Sources
Newspaper Pravda, Oct 5,1957 - original report about launch of Sputnik, front page of "Pravda", followed by Oct 6 article etc. Note that Русские выиграли соревнование = Russians has won a competition is a translation of american headline, put above excerpts from world's media reports
russian, online - Boris Chertok. Rockets and People, 4 volumes, 1999. Boris Chertok is one of leading rocket scientists in USSR space program. This is the most definitive source about USSR space program.
english - Boris Chertok. Rockets and People, Volume IV: The Moon Race
hi! there's always room for more info, but you'll find lots of previous discussion in this section of the FAQ*
USSR's objectives in space exploration
*see the link on the sidebar or the wiki tab