You said you can ping a URL? Like when you ping www.google.com
you get back 4 replies? If so, then check your firewall, anti-virus, anti-spyware, security applications.
Because if you can ping www.google.com,
it's not a DNS issue. More of a application layer issue.
Check your web browser settings also, might want to do some spyware check or try using Firefox. Get it from someone who has Internet, put it on a CD, install it on your computer.
By the way, what is the model of the modem/router? Might want to check that out too, for it might have Stateful Packet Inspection. Which in many case is the problem.