The "Internet" light flashes whenever there is any activity, which will probably be most of the time. It will also continue to do this even if none of the computers connected to it are running, so this is not in itself the cause of your problem.
<<< I can connect to the internet but then get disconnected when the lights start flashing >>>
In what way are you "connecting" and getting "disconnected"? The router's WAN port should be connected to the Cable Modem and your computer's network adapter connected to one of the router's LAN ports (or "wirelessly"). There should be no additional connection (such as USB) between the PC and the Cable Modem and, if you have any software installed relating to a USB connection, this may have to be uninstalled to avoid a conflict.