Hi Joe F,
Welcome to Vista Forums!
First I'd like to try an experiment. Connect directly from the PC to the Modem (bypassing the router if it is possible to do this given the connectors - if not, skip the rest of this and the next paragraph) and then go to Start / Control Panel / Network and Sharing Center / Setup a connection or Network / Connect to the Internet / Broadband / Fill in the necessary information and then click Connect and see if you can connect. If it works, open your browser and verify it works OK. Then Go to start and shutdown the computer.
Now restart the computer. I'm not sure if it will start automatically or not as we really didn't set that up but it might. Go to Start / Connect To and check the available connections. Your local connection should show but be disconnected since we aren't attached to the router. Look at the broadband connection and see if it is connected or not. If not, then select it and click on Connect. Fill in the username and password if it wants it (it may or may not - we didn't configure that either). See if it connects. If so it connected the first time without rebooting. Reboot a couple of times to confirm that you still don't have the same problem. If the connection continues to work, then we know that the problem is with the router setup or the local connection (if the internet goes through it). Whatever happens (if it works or doesn't work), give me all the details in your next post. If it does not work, then go to Start / All Programs / Accessories / Command Prompt and double-click to open it. Type ipconfig /all and enter. Scroll so all the broadband connection information appears and take a screenshot and attach it to your next post. That should help me understand how it is setup and may show me where the problem is.
Reconnect the computer and modem to the router and reboot. Then go to Start / All Programs / Accessories / Command Prompt and double-click to open it. Type ipconfig /all and enter. Scroll so all the local connection information appears and take a screenshot and attach it to your next post. That should help me understand how it is setup and may show me where the problem is. Make sure you name the two screenshots so I'll know which is which and save them as .jpg files before you attach them.
I hope this helps to identify the cause of the problem. Post back if you have any troubles along the way and I'll try to provide better details.
Thanks and good luck!