How do you prove that a logic system is sound?
I am aware of the fact that a logic system must be sound, in order to be useful.
And there is your answer. I understand “soundness” to mean “that which aligns with observable reality“, or “that which is useful”. You design and conduct tests to show that your system produces results that agree with the things that you and others can see and use: in other words, that your system is practical.
In general,  how much work is required to do so,  how do I go about doing so, and  where can I find some references online and in non format of how to do this.
 Well, a lot of work is necessary, quite frankly. You are building a new system of thought. Such efforts take serious time. This is not at all sarcastic, or intended to be discouraging: expect to spend a lifetime on this project.
 I recommend doing so by studying the work of history’s system builders, such as Plato, Aristotle, Leibniz, Kant, and Marx. What did they do to assure that their ruminations squared with observations? What did they get wrong? Or right?
 I would start with Project Gutenberg. The project has done the heavy lifting when it comes to finding and making public original historical documents. And never forget SEP— the Stanford Encyclopedia of Philosophy.
Good luck to you!