first and foremost you may need to check on the ethernet port of your modem if it's working fine because that is the only way you can hook it up to a router. to check if there is a problem with the ethernet port on the modem is by verifying if the ethernet adapter on the computer is working properly(go to device manager and expand the network adapters group there should be a 10/100 dast ethernet adapter installed) if there's none then that's the reason why the ethernet on the modem fails to give your friend's computer an internet access, know if the ethernet is installed properly, you may need to check on the modem manual if there is an option to disable the ethernet port on the modem maybe that is the reason why only the USB port is working on the modem. after going all through the process then you can now plan of buying a router.
to setup a router depends on the configuration on the DSL modem that verizon provided you but the best configuration for you modem is to set it in bridge mode, connect the DSL modem to the WAN/Internet port on the router that you'll be purchasing and setup PPPoE, using the setup CD that came with the router or manually by gaining access to a setup page.
if your planning to buy a linksys wireles router, to access it's setup page you need a browser(IE, Mozilla, Safari) type on the address bar http://192.168.1.1
if a confirmation box appear, just leave the username blank and type 'admin' for the password now you will have access to the setup page, configure PPPoE the enter the userID and password that verizon provided you for your account.
By know you should a working router through your modem, if you have further question regarding the setup just post a reply.