You can mount one of these next to your phone block in the furnace room, and punch down the wires to something like this:

https://www.amazon.com/dp/B00UVQI8B6

Pick up "X" amount of keystone-style RJ45 jacks for use in each room. Here's a 10-pack:

https://www.amazon.com/dp/B00IO3HEN6

Then, pick up some face-plates, to replace the existing ones.

https://www.amazon.com/dp/B0034XRLGQ

Lastly, a punch-down tool for the punch-down block and wall jacks:

https://www.amazon.com/dp/B0072K1QHM

Both the jacks and the punch-down block have color-codes on the sides for use with two different wiring schemes. Pick one scheme (TIA-568A or TIA-568B) and stick with it throughout the installation.

Here is a cheap testing tool, to make sure that you've wired things correctly. (Plug it in on both ends, turn it on, and make sure that the lights are sequencing identically on both ends.)

https://www.amazon.com/dp/B01M63EMBQ

Easy-peasy.