You may also be interested in this previous question about Soviet reaction to the successful Apollo 11 mission, which was buried deep in our FAQ. Hopefully, some others will chime in here as well.
That would depend of which Soviets you were asking about. For the general public, it was limited, as it was given little coverage (a small notice on an inside page of Pravda, if I recall correctly).
Reaction among cosmonauts was obviously different, given their careers as well as access to much more information. Alexey Leonov, the the first man to walk in space as well as the one in training to be the Soviets' first man on the moon, described the feeling as "white envy" (meaning, he was very happy for the Americans who did it, but jealous that they did it first). He was also upset that Soviet media did not broadcast the landing as was done in the west. I can't say much about the government response, other than that it eventually led to the cancellation of the Soviet lunar landing program.
Source for both of those is Two Sides of the Moon: Our Story of the Cold War Space Race, co-written by Leonov himself along with David Scott, commander of Apollo 15. I recommend it highly, both on it's own merits as well as an interesting view of the Soviet program.
The documentary ["The Planets"](http://en.wikipedia.org/wiki/The_Planets_(TV_miniseries)) sheds a little light on the reaction of Soviet engineers. It's on Netflix. The following quote is from a Soviet engineer (~30 mins into the episode "Moon"), who I believe was involved with Luna 16, the Soviet probe that returned soil samples about a year after Apollo 11:
We were really glad we'd got some lunar soil back at last. The Americans had already brought back many kilos, and we only brought a few grammes. At first, I wondered if it had all been worth it. But then the geochemists told us it didn't matter whether it was kilos or grammes for chemistry, that wasn't important.
The series has a few interviews about various probes and milestones from the Soviet engineering perspective.