We are talking about getting the best gain-before-feedback (and the highest signal-to-noise ratio while we are at it). We are not talking about deliberate choices you can make to alter the sound through less efficient gain staging. So we will set aside: working the mic, lowering sensitivity to get rid of unwanted sounds, and those instruments like Electric Guitar where deliberate manipulation of the gain staging and the resulting distortion is part of  the sound.

Simple Language - General Principles
Let's drop the terminology for a minute and talk about what is going on and what we are trying to do.

Assumptions that will affect you how you approach this

1. Every step in the signal chain is an opportunity to pick up noise. It is probably already there, so you  want to avoid amplifying it.
2. Every time you amplify something in the signal chain, you are amplifying the noise too.

1. Start Strong
   Start with a strong input and avoid amplifying along the way.
   
   
2. Less is More: Mixers and Effects and Gain
   We're talking about mixers and effects processors here. If you can pass the signal through a device without affecting the level, you're probably doing fine. This often looks like a setting of "0"  and is sometimes called unity gain. The idea is that you are neither amplifying nor attenuating (diminishing) the strength of the signal.
   
   
3. Amplify Early
   If you have to boost the signal, do it as early as possible in the signal chain.
   This means, if you have a two places where you can boost the signal, choose the one closest to the source.
   Examples:
   1. Choice: microphone or input trim?
      Preferred: Raise the level of the microphone. Move it closer to the source and if possible, make the source louder.
      
   2. Choice: On a mixer ... input trim or channel strip
      Preferred: Raise the level on the input trim. Note: If it's not clipping, it's not too high.
      
   3. Choice: On a mixer ... channel strip or master output
      Preferred: Raise the level on the channel strip. Note: If you have to get above "0" or unity, go back to the beginning and try to get a stronger signal.
      
   4. Choice: Mixer or Power Amp ... mixer master output or power amp trim
      Preferred: Raise the level with mixer master output.Note: If you have to get above "0" or unity, go back to the beginning and try to get a stronger signal.  
      
      
4. Cut Late
   This is the same as the above described a different way. If at some stage the signal coming in is too strong, turn it down at the receiving end. That is, turn it down at the end farthest away from the source.
   

Say it ah... Gain?

Every time you amplifying the signal, you are amplifying the noise. So start strong, keep the level constant, if you have to... amplify early, and cut late.

Disclaimer:
I have no formal affiliation with Bose Corporation.
My opinions are my own.

As a courtesy to the owners of various trademarks, I try not to use the term Bose PAS when referring to the Bose® Personalized Amplification System™ Family of Products.