There are way too many dialects to adequately cover China, as a whole.
Mandarin is supposed to be the official language, as debodine indicates, but, in any given region, local dialects are often spoken between native speakers in that region, and in many cases, the dialects are so divergent that they might as well be a different foreign language, particularly in the case of local idioms.
That said, again, as debodine indicates, Cantonese is a good second choice for both Guanzhou and Hong Kong. Cantonese is quite divergent from Mandarin. When I was little (~9), my mother worked in a Chinese bank, and everyone there spoke at least Mandarin and Cantonese, plus, perhaps, one other dialect. Some woman born in the rural part of Guanzhou came in, and, oddly, I was the only one in the bank that day that could understand her, and that was only because my sitter spoke that exact dialect, and I picked up a few words.
Shanghai has its own dialect, which is quite unintelligible to a Mandarin or Cantonese speaker, but is closest to Mandarin, and if you listen long and hard enough, you can make out a few words.
Beijing has a regional dialect, but it's reasonably understandable to the average Mandarin speaker.
Naturally, they all use the same written language, so one can always resort to that.
So, end result, you need to determine where you might be doing business, and the choice will boil down to Mandarin, Cantonese, or Shanghai. You can get by with Mandarin, but you might get poor service in some restaurants, but that would be either because you speak Mandarin, or because you're noy Chinese ;-)
As debodine indicated, Hong Kong and its mainland surrounds, Guanzhou, will be well served with Cantonese, but you still might need Mandarin, as there has been lots of transmigrations.
From a strict feasibility perspective, Mandarin and Cantonese are pretty much the only two dialects that you can readily get language lessons for, though...
TTFN
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