The telephone port in your computer is very likely a dial-up modem. You would hook a phone line directly to this, and need a dial-up internet service to use it.
The arris modem may be a dsl modem which you would hook the phone line directly into then hook your computer to the modem using an ethernet cable.
You would need an internet service provider for either of these. They will almost always help you get connected so I would suggest contacting them.