Okay, first of all, completely forget what Verizon told you. It's confusing you.
Second, the HDMI connection from the Roku to the TV has nothing to do with the network connection. HDMI is only for the video signal and sound.
One thing that is not clear in your posts is what kind of router you have, or if you even have a wireless router. If you need help, you will be asked for the make and model number of the router. You might want to find that out first before proceeding with asking for assistance here or on the phone.
How far away your router is from the Roku player will most likely determine which network connection method you choose.
You have two ways to connect the Roku player to your internet connection:
1. Wired- Using a CAT-5 ethernet cable, you connect one end to the router and the other end to the Roku player. In the Roku player on-screen setup, you would choose "wired". If your router is only 10 feet away from the Roku player, this might be the best solution for you.
2. Wireless- Think about the word "wireless". It infers that there is no wire involved. So, there is no physical network cable connection between the router and the Roku player. In the Roku player on-screen setup, you would choose "wireless". In order set this up, you will need to know your network name, also referred to as the SSID, and the wireless security password, if the router had been set up with some kind of security (either WEP, WPA or WPA2). Without knowing the network name or the password, you will not be able to complete wireless setup.
Once you select wireless in the setup flow, you'll see a list of all the available wireless networks around you. Your network name should be in that list. If it is, then select it. If you have wireless security turned on, the next screen will ask for the password. Enter it in using the on-screen keyboard. After entering the password, you will see a connection sequence that hopefully will complete successfully.
If you have no idea what your wireless network name or your password is, you'll need to contact customer support of the company that makes your router. If you need to call Roku technical support for help, they are going to ask you for these two pieces of information, so make sure you get the network name and security password squared away first. Assuming you do, you can call Roku support at 888-600-7658. We also have a support page here: http://support.roku.com/home