Make sure you order a continuity cable tester. It's a $10 item on Amazon and is really something you should have if you ever have to punch down a cable or terminate one with an RJ45 end.
https://www.amazon.com/dp/B01M63EMBQ
Or just to test existing cables that you are having issues with. That $10 investment can save hours of your time.
If you want a basic tester just to indicate if your cables are successfully punched-down, you could use an inexpensive tester, such as this one:
https://www.amazon.com/dp/B01M63EMBQ
It is a sequential cable tester. Plug one half of the unit into each end, turn it on, and check the displays on both units, to make sure they are sequencing 1 through 8 . . . The base unit (with the power switch) will sequence 1 through 8, indicating that the wires have conductivity. The remote unit is the important one. . . if it is not sequencing 1-8, then you have mis-wired something at one (or more) of the jacks.
There are more advanced testers out there that will give you more information, but if you're careful to match up the color-coded wires correctly, you won't need all the bells-and-whistles.
Punch down your jacks on each end to either the TIA-568A or -568B scheme, and you should be OK. It doesn't matter which scheme you use, as long as you use the same one everywhere. Don't mix-and-match.
https://www.amazon.com/dp/B01M63EMBQ/ref=cm_sw_r_cp_apa_glt_fabc_0BGEDE9VFG5VK0XFNE0H
That thing? You can get one similar in Networking tool kits.
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.