When things work fine - run ipconfig
Then when 'New network" is detected - run ipconfig again
Any differences might help answer.
To run ipconfig:
Start, Run, type in 'ipconfig /all' - skip the quotes, don't skip space before /
Look at the results. Compare. Post here, if needed.
Is this on WindowsXP?
Is it wired or wireless? If wireless, you might be linking to some other network - your neighbor's
When it says "New private..." does it give an IP address, if so which? If it's like 169.254.0.0 it's no connection, dead ip before you obtain a real IP.
You configured the browser (which) to use router as the DNS server. However, from what you describe , the router uses OpenDNS, perhaps you could tell the same thing, OpenDNS, to the browser.
But, I would, instead, do it in the network properties for TCP/IP to be effective overall, for all communication.