What is the process of putting together a weather prediction?
"Prediction" is very imprecise. From looking out of the window to trying to estimate temperatures or precipitation fifty years from now, it always depends on as much knowledge of the atmosphere-earth interface as possible. That means it always has a considerable amount of built-in error that increases the further out a "prediction" is made.
So, the process of making weather predictions is best summed up as the science of meteorology. It never stops, it is never enough, but if you don't have it, you are only guessing.
"Meteorology is the scientific study of the atmosphere that focuses on weather processes and forecasting."