This is how the bottle is labeled.
3M Liquid Polish - Clearcoat Safe
Part# - 051131-05993
1 U.S. quart
I should say though that this bottle was purchased quite a few years ago, so it's possible this exact product/formulation may no longer be available. If not, I suppose a paint supplier may be able to cross reference the number to something similar that is currently available.
As far as washing beforehand, I've heard the same thing about using liquid dish soap, and think I may have even used it in the past, but I've used a few different car wash products over the years, so I don't know that anything specific is really necessary as long as the surface is clean of anything that might act as an abrasive. And if old wax is more thoroughly removed before polishing, less polish would probably be necessary too.
As far as polishers go, it would depend on which vehicles you plan to use it on. I used to use a 9 or 10 inch Craftsman low-speed orbital because my other vehicle has body panels with larger surface areas, but I picked up a cheap 6 inch low-speed orbital that worked better on the VX because the only "larger" body panel areas were the roof, the doors, and sections of the rear quarters. Neither polisher was anything special either. The 10 inch Craftsman probably cost about $40, and the six inch was around $25. Whichever one you choose though, just make sure that foam pads are available in that size, because I've also tried, cotton, terry cloth, and microfiber pads, and foam always yielded the best results in my opinion.
While this isn't the exact one I got last time (the Titan 6"), it was something similar, and like I said, nothing really special.
Autozone buffer
Waxes? I've used Mequires, Blitz, and even tried the Turtle Wax Ice stuff for awhile. The only thing I've learned is that it's always the polishing process that's made the most difference, and paste wax seems to last a little longer than the liquid stuff. Aside from that, it's just personal preference IMO.