First, congratulations!
You will need to send a lot more than just your marriage certificate. If you search for "documentation" on this forum you will find some posts about supporting evidence that needs to accompany your visa application. You will need to prove not only that you're married, but provide details about your relationship, your financial situation (his and yours), your living arrangements and also your employment.
You don't have to change your name to get your visa, but your visa will be issued in whatever name is on your passport. I opted to change my name in the states before doing my visa application because I didn't want the hassle of having to get to the American embassy to do it in the UK later.
To change the name on your passport, I think you need to do a new application (unless your current passport is less than a year old, in which case they will amend it for free) and you will need to provide your marriage certificate. I changed my driver's license ahead of time, too, and that seemed to be helpful. In IL, I just needed to take my marriage certificate to the DMV and they did it lickety split. To change your Social Security card, there is a form on their site you can mail in.
Hope that gets you started!