To answer your question:For external IP lookup, you would need to access the internet and navigate to a page like
http://whatismyip.com/For internal IP lookup (if you are on a network), you can run
ipconfig from the command prompt.
About the connection issue:I could easily be wrong about this, but I suspect that the home page is loaded from cache, and your not really connected to the internet.
Whats the make/model of the laptop?
What is your connection? ADSL wired/wireless, dialup, etc?
Does CTRL-ALT-DELETE work at all, ever?
If you bring up a command prompt in a stable state and type
ipconfig /all, what are the results?
If you have a router, are there security settings that you need to authenticate with your laptop (eg MAC address filtering)?
Is there security software installed that might it be stopping you from accessing the network for some reason? Try disabling it
temporarily and see if you can connect.
When you "ping the system", what exactly do you type?