I have always preferred to use ethernet cables for my computers; wireless gives me problems. For example, I just got FIOS and I can't get a wireless connection roughly 30-40 feet from the modem. I have the modem in my basement and the wireless connection putters out when I go to the first floor (it's maybe 20 feet from the modem). My question is, can I run two modems in my house using the FIOS internet connection? One in the basement just for my computer, and the wireless router upstairs so little things like walls don't interfere (according to the Verizon website walls are the reason my wireless might not work). I tried to hook up my old cable modem, but I get no connectivity. Is this due to a different signal type (cable vs FIOS) or is it due to something else?
First off having installed many wireless systems for corporations and home installations, wireless signals work best from the top down and not bottom up. So when you say the wireless signal doesn't work well in on the first floor of your home its no surprise. Many things interfere with wireless signals such as microwaves, cordless phones, even bluetooth devices. Plaster walls degrade a wireless signal and librarys simply absorb the signal. What I am getting at is there are lots of factors to take into consideration when using wireless.
Now you mention using your old cable modem. Are you referring to the cable modem itself or is it a router? What brand is it? Does it take in ethernet or a cable wire? FIOS and cable service are different services.
Wireless is susceptable to interference. If you don't want that factor you can go to ethernet. You can ethernet direct to your computer or you can go powerline ethernet, although powerline ethernet may be a bit slower.
As to wireless connectivity, try putting the verizon connection wizard on your computer. Also, it might help to change your router's broadcast channel to channel 11. There are also wireless boosters you can check out. There is also the powerline wireless possibility, which puts another anteanna in any other room in your house.
As to putting your other router on the Verizon network, well, it can be done IF IT IS NOT A MODEM. But adding another router will mean administrative overhead for you, so you will need to be prepared for that. A wireless booster or access point might be better than another router.