Okay, I've worked with VOIP, dad has VOIP, etc...
VOIP stands for Voice Over Internet Protocol. It means that you use your network/internet for phone service, using data, not POTS (Plain Old Telephone System).
However, there are many ways to do this. If you are going with a provider, they provide the link to POTS so you can call other people's cell phones and such. They're the telephone company. If you bought the right equipment, you could have VOIP phones in the house/business then transition to a POTS system to reach the rest of the world, but you're doing the opposite, by the sound of it.
At my work, we got VOIP phones that you plugged a network cable into. They were expensive and highly capable devices that we didn't really use any of the advanced features of. The advantage of this is that you only need to maintain 1 network of wiring.
It sounds like what you want is to utilize your traditional phones with the transition to VOIP happening at your service entrance.
Given the description, you will need TWO adapters, as long as they are the right ones.
For line one, you need a voice line adapter that you plug a phone into. They might be trying to sell you an individual line for every phone. It's also a lot simpler to have an adapter next to your router that you plug ONE network cable into, then ONE POTS cable to your wireless telephone base station. However, you can plug the POTS line into the wall instead, and much like a cheater cable for a power generator going into the dryer outlet, it will provide phone service to the rest of the house, with many of the same caveats: 1. You have to have the phone lines disconnected from the telephone company. 2. It may not have enough power to ring every phone in the house at full volume.
Really, the easiest way is to simply get the device and test it.
For line 2, you need an adapter specifically for the fax system, assuming you still need it. You may also want to consider dropping the second line and simply engaging with a "fax to email" or similar service. No need to print faxes automatically, right? Just get an email with the PDF... There's similar systems for sending faxes.