You just need to wear the amulet of Mara and talk to people that are possible marriage options. Some of theses are Muiri, Ysolda, Aela. You need to complete certain quests before you can marry them though.
Go to UESP for a full list the 3 I mentioned a just a few
Muiri for example:
Spoiler You need to complete the DB quest that she is included in
If you are on PC you can marry more people using console commands