Trust me, it's the router. Find a long piece of Ethernet cable, wire it directly to the modem, bypass the router, prove it to yourself. Then enjoy the wonderful world of networking--and the concept that prior to this purchase you were lucky not to have to change a thing on the router.
All of us on this forum have been where you are, remember that before you go stomping around the room angry because you have encountered a problem you cannot understand.
If you start from the position that it is probably user error, and that there is some stupid box that needs to be checked or unchecked on some freaking hidden menu somewhere, you'll find LOADS of help here in the forums!
Take a deep breath, and power cycle EVERYTHING!