If you try to get your router to work with the router supplied by and set up by Verizon, it won't work well. Or you may succeed in putting Verizon’s router into “bridge mode” so you can use your router properly, but you won’t see the on-screen program guide on your TVs. Here is the right way.
You will need an Ethernet router that in addition to a WAN Ethernet jack has at least one LAN Ethernet jack.
Get Verizon to send someone out. Most of what follows will be done by the Verizon FiOS installer person. Our guy knew about the below, but their phone support people knew nothing about it, or so they said.
The Verizon person must bring along the following box (or equivalent):
Network Interface Module (NIM)
Part Number 519417-004-00
The NIM box will be used (instead of a Verizon-supplied router) to send TV channel information via coax to the DVRs and TV converter boxes.
(As of 2009-05-29 in Florida, at least, Verizon stopped ordering these boxes and stopped offering them to customers as an option, but they still have some in stock. If enough people demand them, I bet they'll start ordering them again.)
The Optical Network Terminal (ONT) is the box that brings optical fiber from the street to the house. It is typically on the outside wall of the house.
Hook a coax cable from the ONT (Optical Network Terminal) to an ordinary passive 2-way coax splitter.
Hook one of the splitter coax outputs to the NIM.
Hook the other splitter coax output (via further splitters) to the DVRs and TV converter boxes in the house.
Hook an Ethernet cable from the ONT to the router's WAN jack.
Hook an Ethernet cable from one of the router's LAN jacks to the NIM.
Verizon must configure the ONT so it passes Internet connectivity through the ONT’s Ethernet jack.
Use Airport Utility to set up the Apple router for an ordinary DHCP connection to the Verizon Internet service. Or if you have a non-Apple router, use the router's web browser setup interface or whatever.