The quickest way to get an engagement ring is to buy it in the Shop with AP. However there are ways to get it free. They are often a random prize in some quest pack items you get for doing a seasonal daily task. Look at the prize pack to see what items are possible prizes.
After you get an engagement ring (item in your inventory) you need to find a character to propose to. For now you may only propose to a character who's gender is opposite your own. Also propose to someone you know and you know will accept or agree ahead of time to accept. When you propose to someone the ring will disappear whether if they accept or decline.
If they accept you are now half way to married. You then need to have at least 2000 coins (normal price) to hold a wedding at the Wedding Chapel. After the wedding ceremony both partners will get an item called a wedding ring which can be placed in the wedding ring slot.