Short answer, a very much Yes, though they started in earnest 2-3 years after the US. The Soyuz spacecraft was to have been the CSM in a similar flight profile to Apollo, with a one man Lander and the whole lot would have been lofted by an N-1 Booster. N-1 booster was cancelled after 4 launch failures due to 1st stage engine faults (it had 30 of them). The biggest issue being turbo-pump failures which were one the verge of being fixed when the program was canned. The biggest issue, however for the soviet effort was the USSR didn't have the cash to fund it to the extent that the USA did, plus there was a lot of in-fighting within the different Soviet design bureaus which got in the way.
Boris Chertok four books are very much worth a read. The 4th book covers the Soviet manned moon flight effort, plus puts it in context with how the US did Apollo. In his view the USA were very communist in carrying out that national effort while the Soviet lunar effort on the other hand were what you would expect from bunch of rival feudal fiefdoms.
Chertok books cover everything that is on this program in a lot more detail, including why Soyuz 1's parachutes failed, What caused Vostok 1's re-entry to go wrong, Etc.
As for this program, loved seeing the footage of R-7's doing failures like the well seen footage of Atlas / Thor / Titan failures, Chertok second book covers all of those shown on this program in detail as well