There is quite a bit to type but I will try to summarize it for you.
In order to do anything, you will first need a Chaperone. This is a person who will do all the dealings for the couple in gettikng you a time to have your wedding and paying for it (you of course give that person the money, unless they REALLY want to see you get married lol).
This chaperone submits an application on your behalf with the time and place you would like to have your wedding and SE will then reply to them whether you got your time or not.
If you did, then YAY you're getting married! If you didnt then you will have to apply again with a different time.
Exactly how the wedding works i'm not sure because I've never seen one yet. But a GM should contact you near the time and get you and your partner ready. Then the ceremony is performed and TADA, you're married.
As far as payments go: I know the wedding dress ensemble is 50k alone, and that is an option on the application. You also have the option of having initials placed on the rings.
The other prices for the wedding in addition to the 50k for the dress is dependant on where you choose to have the wedding and then how many people you are planning on inviting. All this info has top be submitted to SE on your application.
Windy, sandy, and bastok, are relatively cheap whereas if you choose to get married in jeuno then it's really gonna cost you.
As far as any special abilities or such after getting married? Nothing.
It's for show and meaning only. The bride gets to keep the wedding dress and you each have a ring you can equip/unequip at your leisure. You can't share mog houses, you can't warp to eachother at any location (all rumors passed around at some point or another). It's just to be able to say "Hey, Im married!"
I think I covered most of it to the best of my knowledge. You can check out the Playonline website if you want more info. I think there is a whole section on the rules and procedures for getting married.
Hope this helped a little bit at least^^