You don't NEED an internet start up disk. I set up my online without a disk. I just connected the ps2 to the router, and selected online play and it set up itself.
As long as when you connect to the internet it doesn't have the dial tone and the GSSSSSSSSSSHKKKKKKKKK you're ok on the internet connection.
A network cable is also known as an ethernet cable. It is the cable that goes from your internet modem to your router/computer. If you do not have a router, then I suggest getting one. Because without a router, then you can't use the same internet source on more than one device.
Here is a 100' ethernet cable for $50
Cables To Go -
They have different sizes, obviously smaller is less expensive. How long of a cable you need is up to you. Will you be using it for other uses besides the ps2? Also, if you have a TV in the same room as your router, consider moving the ps2 into that room, so you can buy a shorter cord and save some money.
As for routers, here is a link to Best Buy. They have 5 pages of routers for you to choose from to decide which is right for you.
BestBuy.com - Search Results
Hope I could help.