If I connect my Audioengine speakers to a computer, smartphone, tablet, or other music player, what volume level should I set on the player and speakers and what is the ideal overall volume level on the music source to get the cleanest distortion-free sound?

Every setup is different, so some experimentation will be necessary to find your preferred settings, but putting the volume knob around 11 or 12 o'clock is a good starting place if you plan on using your music source (smartphone, computer, TV, etc.) as the master volume control.