How to best separate my signal into frequency regions?
Computational effort is probably totally irrelevant considering the low maximum frequency, implying a low required sampling rate, and the fact that this all isn't done in real-time but after the fact (and computers are fast). So, just use your favourite math toolkit (sounds like you'd like to use Matlab, Octave, or Python/Jupyter Notebooks/Scipy), and design five bandpass filters with the characteristics you need, and apply those.

About these characteristics: You don't give us any info about what you want to do with that signal, what that signal is or applicable information about how flat you need your passband to be, how attenuated your stopband or how wide your transition between. So, that's up to you to figure out. I don't think this is out of reach for you – these frequency ranges sound suspiciously much like EEG, so, maybe, read another five pages in a book about such signals.

Now, no, inherently, the grid frequency doesn't have anything to do with the sampling rate. In the presence of strong narrowband (as in: single-tone) interference, it can be advantageous to use a sampling rate that puts intermodulation products into irrelevant bands, but a) you don't have irrelevant bands and b) considering the low highest frequency, you'll probably significantly oversample, anyway, so that this will be inherently be minimized. There's more than just clock/interferer intermodulation, but that'd be a very hardware-specific aspect, and can't be analyzed with much more detailed knowledge about the hardware you're using.

That grid frequency was given to your for another reason. I'll avoid speculating on this here - because honestly, for a "last year exam in DSP", your questions are a bit too basic, and I don't want to take the excitement of getting things right yourself from you.

So, sampling rate: 2·500 Hz is the theoretical lower limit. You can't use that, because your highest filter band needs some transition width. Not that it matters - any cheap soundcard can do 48 kHz or more, and I'd assume whatever device you're using for sampling isn't worse. So: sample first, then reduce bandwidth digitally with decimating filters. 

Oversampling (that means, sampling with a rate factors higher than your necessary rate) has a dynamic range and SNR advantage, anyway, so you'd do that in a real system if possible.