When should I calculate PSD instead of plain FFT magnitude spectrum?
Now, I would like to show what frequencies the speech has. However,
  I'm not sure what would be the best way to do that. It seems sometimes
  one calculates the absolute value of a Fourier transform, and
  sometimes power spectral density.


If you want to attach physical meaning to your analysis, then go with the power spectral density, (PSD). This is because this will simply give you the power of your signal, in each frequency band. On the other hand if you do not want/care about a physical meaning, but want to know how the fourier amplitudes of each band vary relative to each other, you can stick to absolute magnitude. 

In practice, you can compute the PSD as simply the absolute magnitude of the fourier transform squared. For example, if your signal is $x[n]$, and its DFT is $X(f)$, then the absolute magnitude of the DFT is $|X(f)|$, while the PSD is $|X(f)|^2$. 


  If I understand correctly, the latter works so that I divide my signal
  into parts, do FFT part-by-part and somehow sum these. Window
  functions are somehow involved. Can you clarify this a bit for me? I'm
  new to DSP.


No, this is not true. What you are talking about here refers to the Short Time Fourier Transform, (STFT). This is simply chopping up your time domain signal, widowing it, and then taking the fourier trnasform. At the end of the day though, you will still have a complex matrix. If you choose to take its absolute magnitude, you will have an absolute magnitude fourier transform matrix. If you take its absolute magnitude squared, you will have a power spectral density matrix.