I think I may have mislead you some. The right click is the wrong diagnostic.
Try Start->Network then pick Network & Sharing Center off the lower toolbar.
First check if there are all green lines between the depicted devices.
Any big red Xs mean you have to fix the external hardware.
Vista is quite happy saying its connected to a wireless router.
Unfortunately, the router doesn't have to be plugged into anything else or have an actual internet connection.
IE under Vista uses DNS to determine if it is connected to a Service provider.
Thus if you put in a bad address or any other reason it cannot resolve the DNS name then you get a "not connected" from IE function while Vista still sits there and says its connected.
If all the lines are green and no Xs then
From the Network & Sharing Center you can pick Manage Networks or Manage Wireless Networks.
Only way I know of to reset is delete everything in both categories then create a new connection.
Also get SP1 and the rest of updates for vista, If you havn't already.
Updating seemed to help some with switching to different networks, but it still does a lot of annoying things.
You could also try "View Status"
Under Details see if there is a DNS and DHCP entry.
As a last resort go to properties and uncheck IPv6, some people say this fixes things, but I havn't had to try it yet, and Reboot.
When I get some free time I'm getting rid of Vista on the laptop I have.