Is it better to have a seperate amp for my component speakers and a mono amp for my sub or can I just get one powerful multichannel amp and save some money?
Different solutions entail different compromises. In theory it's better to have separate amps and processor. You get better separation, less crosstalk, lower noise in the processor. However, if you have all the parts built into the one chassis, you have much shorter signal paths, less variation due to interconnect changes, one unified earth rather than the various earths of the separates solution.
In theory, if you hook up all the separates to a good power supply close enough to each other to guarantee the same earth potential, and use appropriate interconnects, you should be better off, but it's not a given for the reasons I give above. There is also the budget question. If you spend the same money on the separates solution as you have for the all-in-one amp, you have paid for several casings, several boxes, several lots of marketing. With only one unit, you've made savings in this regard, which you have spent on the innards. Now, if a low budget is in order, the boxes/marketing are a significant proportion of your cost. If this is a big budget, they're not.
If you have a passive subwoofer, it is better to have a separate amp for it, as subwoofers (in general) need a lot of power to perform their best, and your system speakers perform better without having to share their power supply. That is why it is best to buy a self-powered subwoofer.