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.
You're right, using sockets in C has a difficult syntax. Later languages like Java and Python make it a snap by comparison. The best tutorial I've found for doing socket programming in C is Beej's Guide to Network Programming. I recommend you start at the beginning to get a good overview, but if you just need to get some code working now, you can skip ahead to the section titled Client-Server Background.
Good luck!