The existing answers provide a wealth of interesting information. But the questioner is a beginner and so needs a simple guide to talking about marriage and weddings. Here is my attempt:
The Most Important Words About Weddings
брак--the institution of marriage; a specific marriage
вступить в брак--to get married; literally "to enter into marriage", "to enter into matrimony"
ЗАГС--government office where civil marriage ceremonies are performed
свадьба--the ceremonies and festivities of the wedding day, especially the wedding feast (reception)
жениться--(of a man) to take a wife; (of a couple) to get married
выйти замуж--(of a woman) to get married; literally, to leave home to join a husband's household
Examples
Я приглашен на свадьбу. I am invited to a wedding.
У нас не было свадьбы. We did not have a wedding reception.
Анна вышла замуж. Anna got married.
Иван женился. Ivan got married.
Иван и Анна женились. Ivan and Anna got married.
Как вы относитесь в сексу до брака/до свадьбы? What do you think about premarital sex?
The Parties in Courtship and Marriage
жених--a suitor, an betrothed man, a bridegroom
невеста--a betrothed woman, a bride
супруг--a spouse (male)
супруга--a spouse (female)
спутник жизни--a spouse of either sex; literally a life companion
More Words about Marriage
брачный союз--the marriage union
гражданский брак--strictly speaking: a marriage entered into in a civil ceremony; informally: living together without marriage
бракосочетание--the marriage ceremony
венчание--a church marriage ceremony
свадебный пир--the marriage feast, the wedding reception
замужество--the state of a woman who is married
женитьба--a man's or a couple's wedding; seems to be used to talk about the change in status rather than the festivities
брачная ночь--the wedding night
Words About Matchmaking and Engagement to Marry
сваха--a matchmaker
сводничество--the act of matchmaking (take care since this can also refer to procuring of prostitutes)
сватать--to recommend a match to someone
свататься--to make an offer of marriage
сватовство--the act of proposing marriage
сделать предложение--to propose something (such as marriage)
предложить руку и сердце--to offer heart and hand (made by a man to a woman)
предложение руки и сердца---the offer of heart and hand
просить руки--to ask for a woman's hand in marriage (request made to parents)
помолвка--an engagement
вступать в/заключать брак
can be used. But when and if same sex marriages become the norm in these post-Soviet territories, all these words will come in very handy and start being used both waysон вышел замуж
,она женилась
allowing differentiation based on sexual identity and thus satisfying the need in transmission more nuanced information