I recently made the switch from Comcast cable to Verizon's FiOS service for my home Internet needs. For Comcast my Linksys BEFCMU10 cable modem worked fine for seven years. I don't think you can even buy one of those new anymore. However, Verizon foisted a different wireless router off onto me, the ActionTec MI424WR.
A generic home network router will have five Ethernet ports on it. One port will be for the WAN while the other four are for the LAN. In this case we'll have two such routers. We want our router to be the main one with the ActionTec behind it. Because routers are layer 3 devices we'll need to set up two different IP networks. These network addresses can be anything not publicly routable. For this example our two networks are 192.168.0.0/24 and 192.168.1.0/24.
We'll call our two routers A and B. Router A will be the one that connects to the ISP. The WAN address on A is determined by the ISP (usually via DHCP) so we can ignore that. The LAN addresses on A will be given the 192.168.0.0/24 range. Router B's WAN port will connect to one of the LAN ports on A so it will have to be something like 192.168.0.3. The LAN ports on B belong to the second network so their range is 192.168.1.0/24.
In this example our settop box only talks to router B. So the box will likely have an address like 192.168.1.73. This way you can use your own, better router while keeping basic TV functionality with Verizon FiOS.