The New Testament doesn't describe a wedding ceremony specifically, but there are a few things we can learn from. For example John 4:16-18:
"Jesus told her, 'Go, call your husband and come back.' 'I have no husband,' the woman replied. Jesus said to her, 'You are correct to say that you have no husband. In fact, you have had five husbands, and the man you now have is not your husband. You have spoken truthfully.' "
Jesus didn't recognize the man she "now has" as her husband. People who are married are husband and wife.
Marriage ceremonies were undoubtedly different back then from what they are now, but there would've been some kind of formal ritual. We know there were weddings because Jesus attended one in Cana. Early Christians probably followed either Jewish, Greek or Roman customs.
There's an interesting early letter known as the Epistle to Diognetus. It's a letter from an unknown author to someone named Diognetus, explaining the customs and beliefs of the Christians. It's dated to around AD 200, give or take. It's a great window into early Christianity. There's a line in Chapter 5:
"They marry, like everyone else, and they have children, but they do not destroy their offspring."
And just before that:
"The do not, like some, proclaim themselves the advocates of merely human doctrines. Instead, they inhabit both Greek and barbarian cities, however things have fallen to each of them. And it is while following the customs of the natives in clothing, food, and the rest of ordinary life that they display to us their wonderful and admittedly striking way of life."
This seems to show that they followed the customs of the natives in the places where they lived, to the extent that it didn't compromise their faith. The whole letter is quite interesting.
https://www.christian-history.org/letter-to-diognetus.html
So why are their no detailed teachings on weddings? My guess is because weddings back then weren't the elaborate affairs they are today. The probably involved immediate family and didn't involve "clergy" at all. They just followed the normal customs of wherever they lived. But the New Testament teaching is clear, marriage is sacred and sexual immorality will be judged:
"Marriage should be honored by all, and the marriage bed kept pure, for God will judge the adulterer and all the sexually immoral." Hebrews 13:4
I am reminded of the feast of Jacob’s desire to marry Rachael after serving seven years, but instead lay with Leah that night and was married to her, not Rachael.