To get married, you need to date someone first. You can find potential date partners in the Karaoke Bar in Queens, while other contacts can be met and added to your contact list by organizing a Party in the Luxury Hotel in Vegas. While the party is in progress, you can add characters to your contact list by talking to the guests. You can date those persons later by selecting them from your contact list.

You need to fill the progress bar three times to get engaged to your partner (indicated by the wedding ring icon near your progress bar), and for a fourth time to get proposed. After that, you need to wait 24 hours to enter the Wedding Chapel in the Vegas Hotel for the ceremony.

To be able to marry, you need to be on at least level 15, and at the stage of the main story where this option becomes available (The "Ready for your big day?” story needs to be unlocked.)