|That black cable is a phone cable. It's likely a Cat5 or Cat5e spec, but it's typically used for telephone.|
The blocks in the attic are BIX blocks and again, are typically used for phones but can be used for network too. It's impossible to tell if those in the picture are wired for phone or network but judging from the color coding of the wires on the front of the BIX blocks, it looks like phone.
The reason I say that is, if you have a half ounce of brains, and you're going to use BIX blocks for networking, you take some of your left over (and there's always left over) Cat5 or Cat5e cable (like the stuff punched to the back of the BIX blocks) and strip it completely and use those wires, that are correctly color coded, for network connections. For striaght phone connections you only need one pair and typically, blue/not blue, is used. The images you've posted show blue/not blue so I suspect phone but can't say for sure. However, due to the number of blue/not blue wires, it's entirely possible those are network. I couldn't say for sure either way without testing them.
Since the white cables are Cat5, you said I can get a network patch panel and rework them
Yes you can. In fact, I'd recommend you do this as compared to using the existing BIX block. However, before you begin, get yourself a tone generator and confirm the RJ-45's in your 1st picture do indeed terminate at the BIX blocks. If you take the bix out of the panel, you could identify and mark, each white Cat5 cable as to location.
And how do I hook up my internet to it,
You would want to have a switch located next to the Patch Panel (or BIX block if you use that instead). Your Patch Panel Ports would plug into the switch. Your SOHO router would also plug into the switch.
As far as that goes, if you have a SOHO router with 4 LAN ports, and an 8 port switch, you could have 6 usable LAN ports (3 on the switch and 3 on the router as one on each would be used for the cross connect). However, if you have a router with a single LAN port, you plug that into your switch and are left with 3 usable LAN ports. Don't forget, you could also go out and buy an 8 port switch right.