1) Ethernet. If you connect this way then running your own router is trivial, but by using Ethernet you lose some services related to TV.
2) Coax. This is a bit more complicated. You'll still have to power the Verizon router, but everything behind, and including, your router still will be under your control. You need a Coax to Ethernet adapter like this one:
http://www.netgear.com/service-provider/products/powerline-a...
I don't remember all the specific steps I had to take to enable this offhand. I need to go back and write a howto. In any case, The final setup will look like this:
Verizon ONT --[Coax]--> Moca Adapter --[Ethernet]--> Your router.
Moca Adapter --[Coax]--> Verizon router.
One important point is that you must release your DHCP IP on the Verizon router before your router can obtain one.I think you can dispense with the Coax connection to the Verizon router if you don't care about TV capabilities, but if you don't you might as well switch the ONT to use Ethernet instead and avoid this mess.
My main motivation for doing this was that the wireless capabilities of the Verizon-supplied router were terrible, but it has the added bonus of keeping Verizon's prying eyes out of my home network.