I know that the church was persecuted, so did they still play a roll? Did the soviet government get involved at all? Did they try to change any customs? This question is aimed at Russia, but other soviet republics are on the table to.
In 1917, even before the October Revolution that brought the Bolshevik Party into power, the Provisional Government that formed after the Tsar's abdication in February passed laws that were very progressive in social and political matters. For example, divorce was legalized and women were given a great deal of agency to divorce their husbands if they so chose. The Soviet Constitutions only expanded on social issues - marriage was completely 'divorced' from the Russian Orthodox Church, divorce was expanded and made easier to obtain, and abortion was legalized. I can't speak as much for the countryside but in Russian cities marriage and divorce became a very commonplace thing. The Central Asian republics remained much more traditional in social terms although the Communist Party did, after a great struggle and several decades, denormalize and practically eliminate the wearing of veils by Muslim women. Among Muslims, Jews, and other religious minorities change was slower to come than to Russians but as more generations were born and lived entirely under Soviet rule even more conservative groups came into line with the more progressive norm.
This upheaval of the social order certainly did not come without strains on society at large and after 1936 the rights of women in general and divorce and abortion specifically were cut back by Stalin and other more socially conservative members of the Communist Party. There actually is a striking similarity between Soviet and Nazi propaganda aimed at women produced in the late 30's, both emphasize the responsibility of women to birth and raise a number of sons to then give over to the state as soldiers and workers. World War II also relieved a lot of the pressure on the remaining religious institutions in the Soviet Union and the Russian Orthodox Church was even rehabilitated to serve nationalist propaganda. The post-war period would bring repression once more to the church and freedom to women to marry, divorce, and seek abortions. Unfortunately, because of the prevalence of abortion other methods of birth control were not popularized and many married Russian women had to suffer through repeated abortions if they could not support more children.
Marriage and divorce were entirely state affairs at this point and the format of the Soviet wedding became somewhat standardized - the bride and groom would arrive at the appropriate government office while friends and family waited outside, a short (5 minute) ceremony would be performed by a low-level bureaucrat, and the now married couple would emerge to applause and celebration. Grooms still wore tuxedos and brides would be dressed in conventional bridal gowns and the couple would often go from the government office to a limo which would bring them to a restaurant where they would be rejoined by friends and family for a formal dinner, much like you would see after a marriage in the United States, though Soviet society still had its own peculiarities. In the 60's, 70's, and 80's housing was becoming a serious issue in Soviet cities and the expectation that entire families would live in single apartments together left many young couples without the privacy to enjoy each other's intimate company. The ease of marriage and the guarantee of an apartment as a new family led to many short-lived marriages initiated by desperate and, frankly, horny, young adults. The promise of apartments and jobs also sometimes took new couples away from Moscow and Leningrad and to the Soviet cities of Siberia - larger apartments and better paying jobs were offered to anyone who would leave the big two cities for Perm, Ekaterinburg, Ufa, Omsk, Novosibirsk, etc. The Soviet government had invested heavily in developing Siberian cities and industry and only needed to attract the workers and their families to fill the new cities out.
For a comedic take on the social chaos of the 1920's, I highly recommend star Soviet director Leonid Gaidai's Не Может Быт! (It can't be!), specifically the third segment which follows a young man who met a woman on the street and agreed to marry her but, arriving at his own wedding celebration, can't recognize her without her winter coat and hat. https://youtu.be/DZHXRy-FTuM?t=3443 <--this film and many of the other classics of Soviet cinema are available with English subtitles for free on Mosfilm's YouTube channel!
To add onto this, how much did the sexual revolution in the 1920s USSR affect the attitudes and institutions for marriage?