When you apply for your marriage license, you have to put your 'new surname' on the application so they know you're changing your name. If you don't do this, you'll have to go to ourt to get it changed.
Go to the SS website and print out the form to apply for a new/changed SS card.
Fill it out and go to the SS office (or mail it in) with your marriage certificate (that comes in the mail about 2 weeks after your officiant sends it in), your original SS card and your driver's license.
They will change your name on your SS card and send you a new one.