let's step through this. let's forget the wireless at the moment.
The PCs connect OK straight through the modem. So they must be set up to request an IP address through DHCP. They must also be set up for your name/password.
To work with the router, they need the DHCP, but not the log in details. Have you been adjusting the PC settings each time?
Next, forgetting the internet, does the router work OK just as a DHCP server allocating IP addresses. You can see the addresses for the PCs inside the web utility.
If so, then go onto the next step. If not, there is an internal DHCP server problem. Try allocating static addresses and see if that works.
Next step. If the router is working OK as the centre of it's own small network, the next question is whether it is connecting to the internet itself. It should be set up to request an IP address and with your log-in details. The lights should tell you if it is connecting OK. If it is not, check the settings. You should be able to check on the web utility to see if you have obtained an address or not. If so, go onto step 3. If not, try resetting the unit.
If the LAN is working OK, and the router is connecting OK, then the data is not getting from the LAN side of the router to the WAN side of the router. Check all the security/firewall/filter settings. See if you can ping the ISP web server.
That should give us something to work on....