Okey Dokey dmortimer,
Heres what I think may help you.
1. As a previous poster mentioned, the reason you cannot obtain an IP address directly from your cable modem is because your cable service uses MAC address to authenticate. The MAC is a unique number attached to every network device, so it is different on your laptop than on the computer that works on the cable modem. To get around this, you will need a router.
2. I only have experience with D-Link routers (as I did tech-support for D-Link) and Microsoft (I have one at home). If you want a wireless router and not just an ethernet router, I would suggest either the Microsoft MN-500 or the D-Link DI-624. Both are fairly dependable although my particular MN-500 is a little quirky. Setup is usually rather easy and most routers will have some sort of wizard that will walk you through it. For the DI-624, and most other D-Link routers here's what you would need to do:
A: Turn off your cable modem.
B: Plug the router in to the AC power.
C: Connect the computer that is able to get online through the cable modem to one of the ethernet ports on the back of the router, labeled #1 - #4, not the WAN port.
D: Connect your modem to the WAN port on the back of the router.
E: On the first computer, go to Control Panel > Network Connections > Right click your Local Area Connection (Ethernet) and go to Properties.
F: Scroll through the list and select Internet Protocol (TCP/IP) and click Properties.
G: Make sure you have everything set to Obtain Automatically.
H: Click OK to save any changes.
I: Open Internet Explorer and go to "192.168.0.1". This should bring up your router's configuration page.
J: You should be prompted to login. Username is "admin", password is left blank.
K: Now at the home page click on WAN on the left side.
L: Dynamic IP Address should be selected. Click Clone MAC Address in the middle of the screen, then click Apply, and then Continue.
M: Turn on your cable modem and wait for it to connect.
N: Go to the Status page.
O: There should be a WAN IP (Not LAN) address listed. If it is just 0.0.0.0 then click DHCP Renew. If you get a Timeout then turn off both the modem and the router and wait for 5 minutes.
P: Turn your modem back on first, and wait for it to connect. Then turn on the router, and wait for about 20 seconds til the lights stabilize.
R: You should be able to open a new Internet Explorer window and get online.
You can probably skip steps E - H since thats how your online computer is right now.
3: Basically with any router, you need to clone the MAC of the computer that is able to get online. This is usually done by a wizard. DHCP is also usually enabled by default, so you shouldn't have to worry about enabling it. If you cannot get online after hooking your router up to the modem on the WAN port, and plugging your computers into the router, usually turning everything off for 5 minutes will fix it.
Hope I haven't muddled the waters!