I have an audio source with internal amplifier. Should I max out its amplification, or keep it around middle, or as silent as possible, before feeding it to another amplifier?
Each stage must receive the RIGHT level. Drive it until the peaks start overloading, then back off a notch or two. The final amplifier, the one driving the speakers, should be turned fully up - the exception being when the system is grossly over-powered for the size of room, then it is acceptable to turn a 1kW system into a 100W one by turning down the final volume control.
Where every item has an input level meter, or at least an overload light, this is straightforward. Otherwise it can become more of an art, particularly if you're trying to squeeze maximum volume out of a system, or if one component is noisy (not so much of an issue as it used to be).
The term you want to Google for is "Gain Staging". There's lots of advice out there.