How to use existing satellite coaxial cable to carry the internet connection

It looks like this was originally intended for telephone service distribution.

There's a few ways to go about it depending on how neat and tidy you want the job to look.

1. You can pull those blue wires free from the termination block, crimp them with rj-45 heads, and plug them into your switch. Quicker, Easier, Cheapest.
2. You can replace the punchdown block in the picture with a mini patch panel. More work, cost, but nicer finished look. You will then need to also buy a handfull of 6 inch patch cables to connect it up from your switch.

You should trace and label those cables as well so you know where they go.

It's likely that one does not end in your apartment, but is going to the street or a patch panel elsewhere intended to provide phone service. To prevent damage, do not connect this one to your network equipment.

First step is to get an 802.11ac router. You're probably going to want wireless anyway, so just do that now. Live with that until you decide either you are happy with it, or you need to upgrade. And remember: you won't get ac speeds unless both the router and your device are ac.

If you do need upgrade to wired, buy 2-4 small gigabit switches. Terminate all of the wall plates with cat5e rj45 jacks. At the wall plates with 2 jacks, hook up both jacks to a switch. Since you have daisy-chained drops, the switch will bridge the two halves of the chain. Those cost of doing this should be under $100 for switches and wall plate parts. This is a less than ideal network configuration, but since your bottleneck is going to be your internet connection speed, you will likely hit the limits of performance on this network.