I have a new Dell laptop also(Inspiron 8600) and a D-link "Extreme" router that are both 802.11g compliant. I'm usually less than 30' from the AP through 1 wall at a time, (small apartment) and Network Properties says the connection should be 54 kbps, but my downloads often come in just over 100 kbps. I would recommend D-link to anyone, even I could set it up! (I'm no expert, I just read a few reviews, bought the thing, and set it up (I RTFM). The hardest part was the connection properties and choosing the encryption method.)
Edited by skinbag to clarify- The 802.11g standard blows 802.11b away! Besides the 11->54 kbps upgrade there are methods like D-Link's "Extreme" that use 802.11g but are above the standard. My router (DI-624) was sooooo worth the $. If you plan to surf in your room off of Dad's computer, it's money well spent.
I have Charter cable, and never had to call them, so I'm not sure about "calling Cox". -No pun intended.