Statistical weather prediction Assuming I have time evolution of air temperature, air pressure, wind speed, wind direction, relative humidity and cloudiness, how can I predict weather based on this information? 

I know that high air pressure increases chance of more concentrated sunlight and low air pressure increases chance of precipitation. I know that a decrease in temperature increases chance of snow or ice and also of low relative humidity and vice versa. I also know that higher wind speed means higher chance of a storm and that certain directions of wind can cause a change in temperature.

However I don't know what meteorological parameters I could predict with a combination of these variables. How would I predict the weather based on the time evolution of these variables?
You need a model. You could try to use Bayes theorem to build a model based upon conditional probabilities. See this reference from NOAA about weather forecasting: Probability Forecasting - Primer . 

If I were to try to build a Bayesian model for weather forecasting, with the intention of using a limited data set of observations I had collected myself, I would look into existing software frameworks to see if I could use one of these. For example, the Infer.NET framework is available for free, and has several examples which would be a good starting point for learning about probabilistic modeling.