What will happen?Will our atmosphere head to the next gravity dominant body?Will our moon hold a part of our atmosphere or is too weak to hold it?Where all those gases are supposed to go?
The Sun erodes a planetary atmosphere through its solar wind. Presently, the solar wind passes Earth at 250-750 kps, which is roughly ten times the Sun's escape velocity from earth orbit. So atoms and molecules of gas eroded by the solar wind are carried right out of the Solar System into the beginnings of interstellar space.
At that point (ca. 100 AU) the solar wind starts interacting with the interstellar medium (IM) which is a very low density plasma. The Sun is plowing through the IM at a speed around 25 kps, and there's a bow shock ahead of it and a long tail "downstream." The Sun's magnetic field also plays a big role at extreme distance.
It seems likely that some of the solar wind escapes altogether, while some of it piles up out there. (Note that at 100 AU, there's a lot of space and Earth's entire atmosphere and all its water too, wouldn't make a noticeable impact on gas density out there.)
So, best answer is that as Earth's gasses are stripped away, it gets carried out to around 100 AU by the solar wind where it stagnates, some staying there and some leaking off into the IM.