@SEVENJ9 has it summarized well. Do note that you’ll of course need to obtain a certified copy of your marriage certificate before you can get started. Depending on what state you live in, you may have to apply to have a certified copy mailed to you.
A few more hoops you’ll want to jump through after you’ve taken care of the Social Security Administration, the DMV, and the credit cards: passport, auto insurance, homeowners or renters insurance, your employer’s payroll department, any bills or utilities that are in your current name, your mortgage or lease, and then there are your magazine subscriptions, your alumni organizations, and of course any school you might currently be attending.
Also, note that some banks will let you change your name on your accounts, credit cards and policies over the phone, while some will make you come into the bank with your certified copy of your marriage certificate. I was able to change my USAA credit card, checking account, savings account, homeowners insurance and auto insurance over the phone, whereas Bank of America insisted that I come into the bank. You’ll want to look on your bank’s website or call them to be sure.
But first things first, as 7SEVENJ9 said,
0) Certified copy of your marriage certificate
1) Social Security Administration
[ETA] @magilnyc: You don’t need to provide your current Social Security card. All you need is the filled-out form requesting a new card, which is available online or at the SSA office, and your certified copy of your marriage certificate. When I did it, they didn’t even ask me to provide a current form of identification with my old name on it. (Though the procedure states you’re supposed to provide proof of age, identity and citizenship, so you’ll want to bring your current driver’s license or passport.)