Which methods are most accurate in making forcasts?If weather was 64 degrees Fahrenheit and overcast, There is a storm moving 500 miles north,...

More information would be needed to give an appropriate forecast 3 days in advance.  Meteorology often gets a bad rap for being so inaccurate because most people do not understand the intricacy and many geological influences at play in these scenarios. 

In the above scenario, 64 degrees F is of little help if the geographical location is not known.  In southern FL this temperature could indicate a possible cold front moving through the area while the opposite would be true for Ontario, Canada. 

However, the fact that the sky is overscast indicates there is a low front currently over the area which is always a solid indicator of possible precipitation.  Precipitation cannot occur in a high pressure area because the atmosphere is too 'heavy' to allow water vapor to condense in to clouds. 

The speed of the storm is also way too high for authentic conditions.  The average speed of a moving thunderstorm is about 20 mph.  As you can see, a thunderstorm moving at a rate of 250 mph is unheard of.  Also, if the storm is only 500 miles away and is moving at a rate of 250 mph, the storm will move past you in about 2 hours which does little to help us predict the weather in that area 3 days in the future.  A speed of 250 mph sounds more like the speed of the jet stream (which is another factor that affects the direction of storm, front, and air mass movement as the jet stream moves during different seasons).   

If we remove the thuderstorm's speed from the equation we can then make a more accurate prediction of the weather conditions in the area in the immediate future.  We know for sure there is a low pressure system in the area and a warm front is moving in.  The current temperature is 64 degress F which is neither exceptionally cool nor exceptionally warm.  In absence of other information (location, geographical landforms, proximity to urban areas, etc) it is perfectly reasonable to assume that the area is going to experience at least some form of precipitation, most likely rain, for at least a day or so.  How long this precipitation will last depends on other factors such as the intensity of the current low pressure system and the possibility of other air masses. 

The most accurate method of predicting weather is to consider the stability of the atmosphere in the location being considered.  Low pressure systems, warm and cold fronts, ocean currents, cloud cover, land forms, buildings, and much more all figure in to the equation.  Below are some excellent links that might help you along the way.