From what you have posted and others have contributed, it sounds like you have a SIM Card, Activation, or DOA issue.
Suggestions:
First, if you purchased it from a local AT&T store, bring it in and have them fix your phone or replace it, best way to go.
Second, if you ordered it online (Wirefly.com for example), call their tech line and have them walk you through the fix and if they cannot fix it, they will send you out a new phone, especially if your within the first 30 days of purchase, which it sounds like you are.
After you can advise us on those, we can further assist you, but here are some thought generators for you:
SIM CARD:
1. When you picked up the phone locally, did they install your SIM card, or did they give you a SIM card for you to install when you got it home?
2. When you ordered the phone from the internet, did you purchase it from an individual on eBay or from a certified AT&T seller, such as Wirefly.com (If you purchased it from eBay, state so now so we know what advice to render as this is fish of a different color)? When you received the phone from the shipper, was the SIM card already installed, or did they ship you a SIM card for you to install at home (Normally, when a certified AT&T seller sends you a phone, they install the SIM card for you, because they must activate the phone for you prior to shipping. The only way they can do that is to have the SIM card installed. This also enables them to verify your phone works when it leaves their door, but also ensures the phone is ready to go when you open the box and start the phone).
PHONE:
When you first purchased it and the first time you ever tried to start the phone, did it start? Has it ever started? Depending on your SIM card answers and these, you may have a DOA phone.
Good luck and let us know how it goes..