First of all, jennerjil, welcome to AndroidForums!
What you're describing is odd. I've seen where you can connect to the router but the router isn't connected to the internet (went through this with my sister recently) but being able to get mail and not use the browser is strange. Could it be a proxy setting on the browser? Does Google Play connect? If you go into Settings/Wifi and select the router, you should get a popup which describes the connection. Does the IP address seem reasonable? Don't post your address but I hope you are familiar enough to tell if it's weird or not. Ex, 127.0.0.1 is local host and would be bad in this case.
Sorry, I'm just guessing here.