Airport Extreme, no internet
To start I'd do a reset to factory settings. This support document tells you how.

From there I would start by then connecting a Mac using Ethernet and seeing if you can get it connected to the 'net. 

First would be to establish that the Extreme is handing out IP addresses via DHCP. When you look in System Preferences > Network the Mac should have an IP address from the Express (probably 10.0.. or 192.168..), not a 169...* self assigned IP.

Then can you ping the Extreme from the Mac. Then can you ping your modem. Finally, can you ping "www.google.com" (which will also show that you have DNS working).

If any of that breaks then check the settings on the Extreme until it does and you should be good.