It looks like whoever set this up was a telephone guy. It is incorrect for networking. Does it even work? You will need to get rid of that punch block and put in a switch and a router.
Are you sure that isn't your telephone wiring? Most installers are using ethernet cable for phones.
Your setup should look like this:
You will need some tools like:RJ45 crimp toolethernet connectorethernet jack (for wall plates if you are going to install in your basement, optional)
A spool of cat5e or cat6 ethernet cable
A router if you dont' already have one (even if you use wired mostly, get a wireless N router for phones and tablets, guests, etc.)
A switch (if you have more connections than the router has ports).
At the end of every one of those blue cat5e cables, you need to use the crimp tool and connector so you can plug it in to the router or switch. Google or youtube can show you how to do this. This
is a diagram of the 2 ways to wire the connector (ignore the crossover diagram). This video should help.
Basically you need to just order the wires correctly, insert the wires one at a time into the connector, then crimp. There are 2 different ways to do it, and actually since you don't know how the other end is set up, you will need to try one way and if it doesn't work, re-do it the other way and it should work. It is easy.
To add network drops for your basement, first run the wire. On the router end, do the same as you did with the rest, plug it into the switch or router. On the wall jack end, install a jack instead of a connector, put it in something like this
and install it in your wall. With mine, and this may be overkill, but I put one of these
behind the wall plate to hold it in the wall securely.
As far as cat5e/cat6 cable, cat6 is more expensive, but cat5e will be good until network speeds exceed 10 gigabit. If you want to future proof, use cat 6, but since your house is already wired with cat 5e, I would just keep using that. It will be decades before cat5e is no longer good enough. Some will debate me on this but most people still aren't even using gigabit yet.