Godavari said:
Actually the proper name of our sun is Sol. As in solar energy. Also, our moon is Luna and our planet is Terra.
Those are Latin names, bud.
This is an interesting topic.
I'd think that if we were to colonize another star system, we'd create an endearing name for it other than "the sun" or whatever language have you. If humans will be calling that system home, it'd have to have a better name than X34 2006 or something. In that case, you'd call your star Alpha Centauri or Centauri or whatever name it is in your language, and our star would be the Sun, whatever language have you. Your system would be the Centurian system, while ours would stay the Solar system.
If your a galactic explorer (stupid in realistic terms but I guess for fiction's sake) or something, I'd say that the correct way to say it would be "we're heading towards PROXY" or "we're heading towards the star."
For moons it's a bit weirder. We already call other large natural satellites moons, so I think it could be a planetary thing? Use moon as a colloquial term for any moon, but if you need to give reference, it'd be "Earth's moon" or "Phobos" or "Triton."