Changing my IP on a wired internet

[DukeSinbad]

I did browse around trying to find an answer to my issue, but to no avail. I have a wired home network. Ipconfig does show the 192.168.x.x stuff, but the site itself shows a different ip. ipconfig/all shows that my lease renews every 24 hours. There are two questions that I have though.
I'm fairly well aware the that the default strategy listed won't have an effect as I'm not on the computer that the router is directly linked to, etc.

Could resetting the modem have any effect on the network setup we have on our home network?

Could using a proxy in firefox also shield my ip over connections outside of firefox like games, etc but I'm guessing the resetting deal would probably be preferable to not create latency.

My ISP is Verizon Fios, with a company provided modem that has both hardwiring and wireless connectability.