I'll explain my wired setup and you can figure out what ought to be.
I have a router and a modem. My cable connects to directly to my cable modem. The ethernet port on the modem then connects to the Internet port on the router. My two PCs then connect to any of the 4 LAN ports on the router.
So it seems to me that your modem-router (which is your residential gateway) output (Internet) needs to connect via Cat 5 to the WAN port (for incoming signal) on your router. I've not seen a wireless router which does not allow for wired connection, but I guess it can happen. FWIW, I have a Trendnet router (wired).
Since you are using a wireless router, the computers can connect either wirelessly or wired (should still have 4 ports for wired connections). I just checked the specs and you do. These are your LAN ports.
If you are going wireless on your computers, each system will need the capability for picking up the wireless signals emitted by the router and will need to be properly configured. I don't do wireless, so I know nothing about this.
If wired, the computers just connect via Cat 5 to the LAN ports of the Trendnet router.
If you received any instructions for setting up your router, follow those.
If I've botched this, someone will correct the instructions