You'll probably need a new SIM card, but AT&T doesn't charge for them. Just bring the phone in, ask for an equipment swap and ask if you need a new SIM. At the same time ask if they can transfer your contacts. (You may have to go to a Device Support Center - if you do, they'll tell you where it is, and give you directions if you need them. Or your local store may also be a DSC. Some AT&T stores without DSSCs have the equipment to transfer contacts, pictures, videos and music, some don't and I've seen a few that have the equipment but no one working at the time knows how to use it. But go in, tell them you need an equipment swap to your new phone and go from there.)
In the future, as B.Diddy said, make sure that any contact you enter is entered as an "Account" contact, so it's synced to Google. Then the next time you get a new Android phone, put in your Google email account and it'll sync your apps and contacts in a couple of minutes. (I can't figure out why they leave SIM and Phone as choices for contacts. That's like having a feed bag on a car, because horses needed them. At one time, your only choice, depending on the phone, was to store the contacts on the SIM or in the phone. It's 2014 now.)