After downloading the Arduino software, go to File > Examples > Ethernet. You'll find a bunch of examples that you can run to test your board. If you have a network in the house, try connecting the Arduino board to one of the ports of the network, and connect to the same network from your computer.
In the example programs, set an appropriate IP address for the Arduino and set the Gateway IP. Make sure to put the MAC address of the Ethernet Shield as well instead of what you have in the example (You'll find a sticker on the Shield with the MAC Address).
When both are connected to the network, use your computer to connect to the Arudino by using a program like Putty and to Telnet the Arudino (put it's IP).
I've used the ChatServer program to test my Arduino board last week. Let me know if you need further help.
-Backspaces