<p>I would like to know the simplest and most effective way to open and write data to a socket in the C programming language for network programming.</p>

Reading and writing from basic sockets is not any harder than reading and writing normal files (just use recv instead of read and send instead if write).  Things get a little trickey when you need to open a socket.  The reason for that is because there are many different ways to communicate using sockets (TCP, UDP, etc).