The connections to set-top-boxes and phone do not change. Run an Ethernet line from the ONT to the router's WAN port. Do not change any other connections. Do not disconnect coax from the router, it is needed for the set-top boxes.
Once this is done, contact Verizon and ask them to switch the WAN port on the ONT from coax to Ethernet. It takes just a few minutes once you start talking to them.
Note that if you've had FiOS for quite some time, you might have an older model ONT that needs to be replaced in order to get gigabit service. Still do the Ethernet switch, as that is needed regardless of ONT model.