You activate the mirror targeting, let's say, [[Those Who Serve]], [[Verdant Automaton]], and [[Naga Vitalist]], in that order. The stack looks like this:
Naga Vitalist activation
Verdant Automaton activation
Those Who Serve activation
Now, we let the first activation resolve. The mirror is now a Naga Vitalist. The two other activations stay on the stack, and the game asks all players if they want to do something. You now get a chance to tap your "Naga Vitalist" for mana. That resolves, and the stack now looks like this.
Verdant Automaton activation
Those Who Serve activation
You now let the second activation resolve. The game again asks all players if they want to do anything. You say yes, I want to activate my "Verdant Automaton"'s "3G: Put a +1/+1 counter on this creature" ability. This ability goes on the stack.
Put a +1/+1 counter on "Verdant Automaton"
Those Who Serve activation
Now, the +1/+1 counter ability resolves (if no one else has anything to do), and if you again have nothing else to do before the Those Who Serve activation, that resolves as well. Net, you've generated one mana of any color your lands could produce, and you have a 3/5 "Those Who Serve" until end of turn. (Then at end of turn you'll have a noncreature mirror with a +1/+1 counter on it that does nothing until the mirror animates again.)