WiFi Shield Sending and Receiving Information

I'm working on a project with an Arduino and WiFi Shield to simulate a thermometer. I have a server set up using sockets which is where the thermometer will obtain its temperature via wireless. I have a working wireless router but what is the best way to set it up for communication between the Arduino and server? The two options I see are:

  1. Set up the router as a typical wireless access point and use the internet to contact the server
  2. Connect the router directly to the server without providing internet access and just communicate via the router directly to the server

Any options I'm missing, advice on which will be best, etc would be greatly appreciated.

I have a server set up using sockets which is where the thermometer will obtain its temperature via wireless.

Looks a bit like you used dial-a-word to assemble this sentence. Would you like to try again?

I have a working wireless router but what is the best way to set it up for communication between the Arduino and server?

It doesn't really matter, from the perspective of the Arduino with wifi shield talking to the router, what the router does with the data that the Arduino sends it or where the router gets the data it sends to the Arduino.