Before modern technology, people predicted weather by looking at the sky and clouds. By studying the cloud types and the weather they portent, as described at the following site, you and your kids can learn to do this too.
The following video describes cold fronts and warm fronts, their movement, and sorts of clouds and probable rain associated with them. It also notes how warm and cold fronts are depicted on maps:
Data regarding air pressure, temperature, precipitation, and wind are constantly collected from thousands of weather monitoring stations throughout the country (and world) and compiled into WEATHER MAPS, that are constantly updated. At the following site, you can pull up a weather map any day from the present back (note “previous day” and “next day” in the top corners).
A great exercise is to correlate changes in your daily weather with notations on the map. For example: It has suddenly turned cooler. Can you correlate that with the passage of a cold front depicted on the map? After this, can you predict coming weather? This is what the weather man on your TV is does, albeit she/he has even more inputs of data to work from.