From my many years on the board (first joined I think in late 2006 or 2007) and a bit from school, what I've learned is forecasting is much more than just reading model output. Obviously forecast models are significant in the forecasting process, but forecasting goes beyond just reading model output.
When someone is forecasting and analyzing models, they should be asking themselves questions in their head and coming to an understanding of what is going on within the model to generate that output. Does the evolution make sense? Is this a realistic solution? Are there any biases the model is known to have and are those biases being reflected here? You also want to fully assess a wide variety of data and output from all levels of the atmosphere.
A lot of focus goes right into surface outputs (QPF, snow maps, precipitation totals, etc.) which I get because we live at the surface, but having a strong understanding of the upper-levels, the pattern in place, how the pieces are evolving and interacting will tell you more about what to expect at the surface than any surface product will. I also think having a strong background in the complex mathematical equations can help too.
You also want to be looking for run-to-run consistency, model-to-model consistency, and recent model performance and having an understanding of this can help a forecaster confidence wise in which model to perhaps rely on more.
Ultimately, it's all about experience and understanding of the models, their strengths, biases, and understanding the overall pattern.