You should probably read this article on taking shots of the Milky Way.
Just to pick out one specific example, they show a shot taken with a Canon T2i ("a much older model" DSLR) and the Canon 18-55mm f/3.5-5.6 (basic kit lens). I set it to 18mm and f/3.5, shot a 30 second exposure at ISO 6400.
Now note this is a DSLR (larger sensor).
The exposure required a long exposure (30 seconds is typically the longest you can do without a Bulb mode on your camera) and very high ISO (ISO 6400). I'd consider that the high ISO seriously compromises what you could do with a P&S, which have small sensors and hence extremely high noise at high ISO.
To lower ISO you need to use a wider aperture (again, not an option for a P&S) and/or use a longer exposure (not really an option for a P&S even with bulb mode, as it increase noise more).
If you really want to do that, consider buying (or borrowing) a used older model DSLR and a basic kit lens (like an 18-55) and read that article to see how to proceed from that.
Now you can try and use the P&S to take the shot, but I'd not expect much and I'd suggest you try a night time shot of the sky in a park or similar before taking you trip to get a feel for what's going to happen.