The bride's name goes first. If you put a monogram on a napkin then her initial goes first, groom's last name initial and then his first name initial. You can flip the initials if it spells something you don't like.
The brides name usually goes first before the wedding, like on the invitations it would be the brides name followed by the grooms. However, after you are married, the husbands name is usually first, such as Mr. & Mrs.