Is it possible to set up the official arduino WiFi shield to send data via TCP/IP?
Yes, the WiFi shield can sent TCP/IP data to an existing network.
If so, are there any example programs/tutorials out there that walk the woefully ignorant through the process?
There are some that come with the WiFi library.
Or, on thin ice here using words I don't understand, would I use the WiFi library to have the Arduino/WiFi act as a client?
Do you want it to get data (as a client does) or to serve data (to clients)?