How To Send Commands Over A Network

I have never programmed a network program before and would like to learn at my own pace. Also I am not asking anyone to write this complete program for me, just bits of code to help me write a complete program. (though if you do code it all I won't complain.) I already tried using internet tutorials but none of them seem to work.I am trying to program a simple form with 4 buttons, and 2 users.The first button is a connectivity button. "Green" for connected, "Red" for not.The second button is a "Connect" Button which will connect to another computer with this program installed/active. (I want it as simple as possible)and the next 2 buttons switch between "1" and "0" depending on user input(Default "0"). the first button is the local user button and the second is the remote user button. When the local user clicks the local button then what he/she changes it to will be displayed at the remote site. The same thing will happen with the remote user.

I know how to place all the buttons where they belong, and to change the local text, but I have no Idea how to connect to a remote program or to pass messages between the two users. Can anyone suggest some Simple code to do this. (something that I will be able to learn and use to make more complex code later on.)