Krassyg,
The supplied routers have a checkered history of causing issues if the local network is at all complicated (beyond a couple computers). I used their modem/router for a while at home and there were constant issues. Power failures were a blessing. Without a power failure I had to reset EVERYTHING from time to time. The day that I went direct to the ONT, all of the monkey business stopped and I'm using an older, low end Linksys router. There have been numerous updates of hardware and firmware from Verizon. I don't think that the latest units are quite as bad, but I don't want to risk using the supplied router.
The ONT (Optical Network Terminal) is the interface between the fiber and the house. The ONT has Coax, two phone lines, and one Ethernet output connections. The default installation keeps the ONT's Ethernet output disabled and connects the home network via the coax connection. In this configuration the FIOS modem/router behaves like the cable modem/router's that you are familiar with. As with most modern cable/modems the COAX is also a bidirectional MOCA connection.
Here is a DSL Reports FAQ article. Without the MOCA connection to your coax distribution, the cable boxes will not have access to the guide data or PVR (Personal Video Recorder) functions.
The most straight forward step-up connects your router's WAN port to a LAN port on the supplied router. (use different LAN subnets for your router and their router) This arrangement gives FIOS support access to their router from their side. Without this access, they will have some difficulty diagnosing their problems and they'll be grumpy. A disadvantage of this arrangement is complications when setting up open ports and similar. If you want to use the ONT's Ethernet connection you'll need to have Verizon support enable it. Some support agents will do this in a blink, others will balk.
An advantage of the direct to ONT connection is that Verizon support will (usually) be reluctant to have the customer reset anything if the customer calls Verizon support before calling you. However, some of the support agents think that they are doing the customer a favor by guiding the customer through pressing every "reset" button that they can find. I recommend placing a discouraging note on the routers.
My preferred arrangement is Ethernet directly to the ONT, my own router, and TIVO boxes for the TV's.
Everyone has horror stories about both FIOS and regular cable. My experience has been that there are less issues with FIOS.