The Virtual tour is good, and I don't know why that wasn't an accepted answer. So I'll throw my answer into the ring. I'll recommend you try the NWP and Forecasting Comet module (it's free if to get an account). That'll probably help more than my answer, which is based around the forecasting classes that I took and my experience as a forecaster in the university's student-run weather service.
From the outside, the forecast process is a very mysterious thing. But in reality it is not. Though, it is highly a subjective task and is subject to some degree of variation per person (hence some people are better forecasters than others). It also takes a lot of time to do it right. Here is my pattern if I want to be rigorous.
- Know where I am forecasting for.
Location is important. What do I know about the climatology. Climatology can give a "first guess" on the range of values that can be reasonably considered. The previous day's weather can also be a good first guess. If I am well acquainted with the area, then this becomes a shorter and shorter step.
- Start with observations.
What is happening now? What is reality saying? The amount of space that is looked at needs to be proportionate to the forecast time.
- Make inferences on the current state and causes of weather features.
What patterns are seen? Was map analysis done? Are there certain areas that are colder than others? Are there places that have clouds and others that don't have clouds? What does the radar say? Why, why why why why? If you know the mechanisms that are generating weather now, then you can understand how they might evolve.
- Examine how the weather models have started.
Before you use a weather model, you should understand it. Garbage in=garbage out, sometimes. How well did the model do today? If it is overpredicting temperature right now, will it continue overpredicting the temperature? Will the errors that occurred upstream yesterday occur today?
- Look at how the weather models progress. Question if it aligns with my knowledge and experience.
Taking a model at face value might work for research purposes (unless you are researching the model itself), but shouldn't be done on a practical level or in a rush. What does Model Output Statistics (MOS) say?
- Choose the right numbers or features.
This is probably the step that requires the least amount of explanation. Though the more intricate the type of forecast, the harder and harder this becomes. Does it actually require numbers, or is there some sort of GIS software (like for hurricane trajectory or lightning forecast)?
This can't be stated enough. You must verify how well you did. Decisions need to be made on how the forecast will be verified. What data sources do you know of for verification? If I could move this up to number 1 and still have this make sense, I would. Because this is what you should start off with. This actually goes part and parcel with starting with observations, since observations are what you start a forecast with. Understand the processes of why your forecast was off. This will serve you in the future and in the next forecast.