Between those two choices, I would go with your cable company's technical department first. They should be able to tell you if the connection from the cable to the computer is in working order or not.
If the cable modem in your home is providing a usable signal, then the Internet connectivity problem may be with your computer. Many times a computer's internal settings need to be changed in order to connect with a new type of modem or router.
If the cable modem connection is working, then you may have to call tech support at Dell. They can walk you through a series of checks to make sure your computer is compatible with the cable modem.
It is possible that your wireless router is defective or malfunctioning, but I would try the cable company and the computer tech support before ordering another one.
The cable company may be able to check the router at the same time they check the cable signal to it.