If your router is wireless, all you need is a Wireless NIC (Network Interface Card), for your desktop. You can install it either into the PCI slot or use a USB port. After the drivers are installed, and you connect to your router, by default the TCP/IP settings are set to dynamic so they should pull an IP from the router and DNS servers as well.
Just call Verizon and ask for data tech support. Don't you think it's better to have them walk you through the set up, than getting a bunch of wrong and confusing answers here?
Verizon support is free for their customers.