If you are ready, I would ask for ethernet install. Just connect the ethernet to the WAN port of the new router. If you have Verizon's TV boxes the coax should be connected also, but the internet will come on the cat 5-6 to your router.
Depending on your equipment, you can figure out how you want the eventual setup to be. Here is a guide to help figure it all out:
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Verizon FiOS FAQ