If you have a set of data that you want to find the trends of, then typically one either uses various curve-fitting techniques (which can range from basic linear-regression style methods to more advanced methods) or time series analysis, yes.
To extrapolate from fitted or observed trends, however, is going to require a mathematical model. Sometimes the model is quite simple: you observe that some data is fit well by an exponentially increasing distribution, for example, so you assume that future data will follow the same trend and be described by the same exponential you fit the past data with. The predictive power of this approach is going to depend on whether or not the conditions under which the past data was observed change. If conditions stay the same, you might expect your fit prediction to remain valid. However, if conditions start to change for some reason or another, then at some point your prediction will cease to be valid.
In such cases, it helps if you build an actual independent mathematical model based on what processes you think will cause the observed, rather than just fitting the data itself (although fitting the data can help you choose values for your model parameters). If you get the processes which drive the data correct, then your model can have much more predictive power than simply fitting a function to the observed data and arguing that the trend should continue. On the other hand, if you get the processes wrong, your model may break down in the future as well.
This said, if your university has courses on mathematical modelling, I would look into those. Courses on statistical analysis of quantitative data are also good to look into. Looking at my own university's undergrad courses, some course titles are "applied regression and design", "time series analysis", "advanced data analysis", "topics in applied statistics", "statistical data management". Some courses in probability theory would also be good.