Connect Arduino via wifi and send information to another arduino (via wifi)

HI all, I'm working on a project where I'd ideally like to have an arduino wirelessly connnected to the internet via wifi sending information over the internet to another arduino wirelessly connected to a different network which receives the information.

So that being said could you help me understand what sort of hardware and software I'd need step by step?
In other words:
Step 1: How should I connect the arduino to the wifi (what hardware and software allow information to wirelessly be transmitted).
Step 2: How could I then transmit this information to someone else's network.
Step 3: And finally have this information be transmitted to an arduino via wifi.

I have zero background in networking (literally, zero) and a small background with Arduino and coding so if you could explain a lot of basic terms as you use them like TCP/IP , or shield for arduinos, that'd be great.

Step 1: How should I connect the arduino to the wifi (what hardware and software allow information to wirelessly be transmitted).
Arduino WiFi Shield: http://arduino.cc/en/Main/ArduinoWiFiShield

Step 2: How could I then transmit this information to someone else's network.
Specify their IP address and port number. See File->Examples->WiFi->WiFiUdpSendReceiveString

Step 3: And finally have this information be transmitted to an arduino via wifi.
Set up the destination router with port forwarding to allow access to the internal Arduino. Connect to the IP address and port number that gets forwarded to the Arduino. See File->Examples->WiFi->WiFiUdpSendReceiveString

Thanks for the insight! Is working with the wifi shield that difficult in your opinion? Is there anything I can do in the meantime (before it ships to me) that way I can better understand how to use it?

I think you also should look at the yun.
That one is made for this type of functionality
Best regards
Jantje

zloz:
Thanks for the insight! Is working with the wifi shield that difficult in your opinion? Is there anything I can do in the meantime (before it ships to me) that way I can better understand how to use it?

If you have a couple of Ethernet shields handy you can practice with hard-wired networks.