The first thing to have handy in this case is what kind of connection your old modem used to connect to Verizon. If it used PPPoE, you will need to ensure that you have the Verizon Username and Password available for your Master Account. Though many areas can use whatever PPPoE Username and Password you'd wish to use, some might still enforce the use of the master account login. If your old modem however used Routed Bridge mode, you're probably in one of Verizon's few remaining DHCP areas. These areas require you to either leave your modem disconnected overnight or require the cloning of your old modem's MAC address in order to work.
Once you've figured out the above, ensure the Motorola modem is obtaining sync (Solid DSL light normally on most modems) and also ensure that the modem is using the VPI/VCI settings of 0/35 respectively with an Encapsulation of LLC for the ATM Virtual Circuit (if it hasn't automatically detected this). Lastly, ensure the Motorola modem is configured in Bridged mode so that your Airport can hold the Public IP.
Next, using the information I first mentioned, determine what you need to configure your Airport Extreme to use. If you are in a DHCP area, your Airport should be left on DHCP to obtain IP address information. If you use PPPoE, your Airport should be set to use PPP/PPPoE with the mentioned login details. From there, once this is saved the Airport should obtain an IP address and serve up the Internet connection.