Is work done by a system negative delta G or positive delta G?
Depends what you mean by that.  I agree it can be a little bewildering to figure out exactly what all these words in thermodynamics mean -- energy and work and heat and free energy and God knows what else.

It may help to remember in all such cases we're just talking about energy.  That's it, that's all there is.  The various types of energy that we call enthalpy or free energy or whatever are just different ways to identify part of the energy with the system and part with the surroundings.  Most of the time, changes in enthalpy or free energy or energy are just statements that the energy got transferred under certain conditions, for example with the system at constant temperature and pressure, or maybe constant pressure and volume.

But the important thing to remember is it's all just energy, and you can use the same intuitions about it you'd use if you just used the word "energy" everywhere (up to a point, of course, because the distinctions aren't meaningless).

So for example in this case, I hope it makes intuitive sense to you that if a system does work in its surroundings (causes weights to be raised, springs to be compressed, batteries to be charged, whatever) that it has to lose energy for that to happen.  Energy has to flow out of the system and do the work.  So that would suggest the energy -- or enthalpy -- or free energy -- has to go down. And so they would, generally.  That's a good intuition to have, a good starting point, although there are certainly more complicated tricky situations where you'd want to refine your first guess.

Usually if you're talking about free energy, you're talking about a system at constant T and P.  For that system, to do work, it has to lose energy, which means G goes down, dG &lt; 0.  On the other hand, if work is done on the system -- springs within the system are compressed, batteries charged, et cetera, then dG > 0 and the free energy goes up.  All just as intuition would suggest for what happens to the energy.