After you are married your marriage license must be returned to the county clerk the issued it. They will then issue you a certified copy of the marriage license proving you are married. Once you have that, you need to change your name through Social Security. You can find more information about that on the social security website here: https://www.ssa.gov/personal-record/change-name
Your name is legally changed through social security. They will issue a new card with your new name - you keep the same number. Then you can change your name on your drivers license. With the new social security card and new drivers license, you will will be able to change your name with everyone else.