DIY weather prediction
People are trying to predict the weather since ancient times, but the first real attempts for numerical forecasting happened in the middle of the 20th century. The results were very unsatisfying and even led to some weather forecasters suicides.
Though the equations that are used in weather forecasting models considered chaotic, there has been a significant development in this field in the past few decades. Mainly due to the advancements made in computers calculations speed.
How to begin
First, in order to be able to predict what will happen in the future, you must have a firm grasp of the meteorological conditions that exist right now. For that we use satellite imagery, rain radars and observations.
There are many weather satellites and they come in 2 types:
1. Geostationary - rotating at 36,000Km above the same spot on the equator all the time.
2. Polar - Orbiting the earth from pole to pole at a height of less than 1,000Km.
The polar satellites take better resolution photos, since they are closer to earth. But weather forecasters usually use the geostationary satellites since they take a picture of the same area every time, and thus making it easier to create a "movie" in which you can see the development of the weather systems.
There are sites offernig free satellite images, like sat24.com or eumetsat. There are many types of images, For example the visible light spectrum or IR spectrum, and you must understand what you see in each one.
Though it may look intuitive, it requires some level of expertise to decipher the images correctly. Much of the learning here comes from experience.
Most countries have rain radars, And a short search on the internet can get you to a rain radar that covers your area.
In those radar images you can see where it is raining right now and to observe the change in the intensity of the rain. This is a very straightforward tool, easy to use and to understand.
Most countries also have a network of weather stations that record the current meteorological state. This data is usually available for free.
Additionally there are always weather-freaks that have weather stations at home and sometimes share the data on the internet.
The examples above are of ways to obtain surface observations. Another type of observations are from upper air. Most modern countries launch at least twice a day a Radiosonde, also known as a weather balloon. If you don't know where to search for this data you can start here.
In the radiosonde data you can see the temperature, humidity and other parameters, all the way from the ground to an average height of 15Km. This data can give you a good assessment of the atmospherical instability.
Start to forecast
The tools I have presented so far can be used for what is called "short-term forecasting". By knowing how the weather developed in the past several hours, you can assess how it will continue to develop in the next hour or two. This can be very useful in many occasions.
But if you want a "long-term forecasting" the tool for you is the weather model. These models numerically solve the 7 equations of the atmosphere, for every time step and for every point on the model map. That means A LOT of calculations, which are done in supercomputers.
The best known models are the GFS (Global Forecast System) and the ECMWF (European Centre for Medium-Range Weather Forecasts), but there are other models that are used by forecasters.
Much of the models data is available online, a good site to start from is the weather underground that has a very convenient interface.
When you move forward in time you can see how the atmospheric parameters change and derive the future weather conditions. Don't forget that the models are sometimes wrong, especially for the far future, so always put some uncertainty in your forecasts.