1. The only thing that matters in engineering is the discrete Fourier transform (DFT). That’s what anyone who wants to calculate anything must use.
2. The fast Fourier transform (FFT) is used to to calculate the DFT. Cue up Matlab or LabView.
3. Knowing the units of the abscissa in sample and frequency space is next.
4. Studying the sampling theorem so hard you can describe it in a number of different contexts, rigorously, is critical.
5. Knowing what aliasing and antialiasing are is critical.
6. Knowing how to use transforms to interpolate is next.
7. Knowing how to mess around with the real and imaginary parts of the complex DFT to improve the S/N is a good achievement.
8. Convolutional filters.
9. Smoothing data sets with non-causal filters.
10. What windowing functions, and why?
Yes. The problem is assuming the only thing that matters with the fourier transform is the DFT. Point #1 is incorrect. Sometimes sampling theorems are misleading (not incorrect). There are cases where it is highly advantageous compute a continuous fourier transform without a DFT.
I think my answer was highly oriented towards handling real data. But even in simulation, any calculations are discrete.
It helps tremendously to understand continuous Fourier transforms, because most of the theorems have immediate discrete analogs.
"The Scientist and Engineer's Guide to Digital Signal Processing" by Steven W Smith available online here: http://www.dspguide.com/pdfbook.htm
One of my colleagues lent me a hardcover copy very good explanation I found it a huge help for a recent project I was working on (Sonic vibration sensor).
Clearest explanation of everything I was able to find. Fourier transforms were covered during my engineering degree but it had been some years since I'd needed to concern myself with them. Top google hits were not really helping and I found this textbook was invaluable.
I recommend Coursera Digital Signal Processing by Martin Vetterli https://www.coursera.org/learn/dsp. He does a good job of explaining the math behind
the DFT. Any finite signal can be expressed as a linear combination of complex exponentials - that was the aha! moment for me.
Have I left anything out?