What you need is a Cisco (or generic) branded Serial Console cable. The Console cable can be made, but that is a hassle. Ask at work if they have an extra they can give you. Plug said cable into the console port on the router. Plug the other end into your computer. Use either Hyperterminal or PuTTY, set to the COM port you are using (generally COM1), and set the port to 9600bps, 8 data bits, no parity, 1 stop bit, no (none) flow control.
Then turn on the cisco router. You should then see a bunch of stuff wizz by.
Eventually you'll come to a prompt. Hopefully the router was reset to defaults, in which case it will ask you if you want to run the configuration tool. I usually select no, and then it should dump you to a command line.
Type "en" or "enable" to get into admin mode. Then from there you can do "sh ru" or "show running-config" to show the current config that is loaded on the router. You can also do "sh ver" or "show version" to see what the router is, hardware in it, IOS version, etc. If you noticed, you do not have to type out the full command, usually just the first few letters which saves on typing.
[ame="http://www.amazon.com/CCNA-Certified-Network-Associate-640-801/dp/0782143911/ref=pd_bbs_9?ie=UTF8&s=books&qid=1196995047&sr=8-9"]Amazon.com: CCNA: Cisco Certified Network Associate Study Guide, 5th Edition (640-801): Books: Todd Lammle[/ame]
That book there is one of the better learn Cisco and learn the CCNA. Note that the CCNA just changed either last month or this month. It used to be either one big exam or two parts. But I forget what it has changed to.
I don't have my CCNA yet, but have been playing around with stuff. I've got two 1600 level routers, one 3640, and five 3662 routers that I've been playing around with. What you'll want to do is get a second router and then some serial DTE-to-DCE cables, so you can connect the routers via their serial ports back to back, so you can set up routing and learn all that jazz.
Hope this helps. PM or e-mail me if you need help. I'm not a master at this stuff yet, but know enough to get my way around.
Edit: Another thing to learn is the OSI layers. The CCNA book I listed has a good couple of chapters on the fundamentals of networking which you really need to know and have down pat before you step into the Cisco arena.