Want to build an ethernet-connected USB or battery powered piezo alarm gizmo -- what's a good kit?
The hardware you suggested sounds right for the job, given your skill level as well.  Are there non-arduino solutions? Yes. Would they be easier to code on?  Probably not.  Given that this is your first venture into electronics, I highly recommend the route that you've already suggested.

You will need the arduino board as well.  That is where your program will go.  The ethernet shield is an add on peripheral.  The software to write the programs ("sketches") is free as well.

Happy hacking! http://www.arduino.cc/en/Reference/Ethernet