Um... do I have to state the obvious?
-He's using wires, so wireless is very much out.
-Assuming the cable modem and the PC are connected by a patch cable and not a crossover, the modem has a DHCP server.
-Going by the language used, he's not that experienced but is looking for an easy guide. Nothing fancy or costly.
So we don't need a router OR a wireless gateway. Geez, people...
My recommendation is to buy a little box called a "switch". You can get an 8-port switch for $20-30, usually. You plug the cable modem into the switch, then get as many network cables as you need, and use them to plug your computers into the switch. Voila, shared internet to more than one person.
Now, your other question. The reason you can't put in another network card and connect it to another computer using a regular cable is pretty simple when you think about it. The cable has different wires for different things, but you can just think of them as SEND and RECEIVE. When you connect two computers together with a REGULAR cable, the send and receive wires don't match up. However, there's another kind of cable called a CROSSOVER cable, which has the wires on one end switched around so that the send and receive wires DO match up. They're "crossed over", which is where the name came from.
The crossover cable is useful for two people sharing the internet, as it's going to be far less costly than a switch, but once you get three or more people it's very inefficient and costly (two cards per PC as opposed to one, apart from the guy on the end of the chain)
NOTE: when i say "plug in" i mean with a regular network cable. Not a crossover.