From the internet you go through several hops (or hoops) to get to your computer. Internet to ISP (cable company) to Modem then to Router (linksys) and then computer. My first impression is that one of your ethernet cables are bad, or the WAN port on your router is bad.
Internet to ISP is good or you have to get a new ISP. ISP to modem sounds good since I assume your modem lights connection comes on. So now you have modem to router to computer or modem to computer.
Since you can can connect modem to computer looks like your computer is set up property. At least to the ISP. First, I would check Computer. Start -> Control Panel -> Network Connections -> Local Area Connection -> TCPIP -> [Properties] then Check to see of you have Automatically Obtain IP Address Checked. This will see if you have a static IP Address from the cable company.
If you have a static IP address from your ISP you will have to setup your Router with the static IP Address. Turn on DHCP from in your router and Automatically Obtain IP Address on your computer.
Now to check the cables you will have to do a process of elimation. Since your you cant really see what's going on with your router, you have to check the router with your computer. Connect the Router to your computer with one of the cables. Now see if you have access to your router from your computer in IE goto something likes 192.168.1.1 and see if you get a login screen. Then unplug router and switch cables. Turn on router and access configuration screen again 192.168.1.1.
If both cables do not work, maybe the WAN port on router is bad. If 1 works and 1 do not try using a different ethernet cable. If both work it could be the static IP to your ISP.
The basic configuration is: Router set up to Obtain IP Address. and DHCP server ON. Computer Automatically Obtain IP Address and no other configurations.
Hope this helps some...