Have you actually "Connected" to the wireless network? In other words, have you run the wireless adapter's software, scanned for "Available Wireless Networks", found and highlighted the one that matches your router's SSID (network name) and selected the option to "Connect" to it? You should be prompted to enter any WEP or WPA encryption key and, once "Connected", the wireless adapter should get its IP address by DHCP from the router and you should have network and internet access.
There is a difference between the Wireless Network Adapter and Router communicating with a good signal and "Connecting" (as in "logging in") to your Wireless network. Note also that, if an incorrect WEP/WPA key is entered, you may appear to have "Connected" to the wireless network, but the connection won't work.
Connecting an 802.11g adapter to an 802.11n isn't in itself a problem as there is backward compatibility.