As a Canadian (very similar to American English) we say it like "grosh-ry" (two syllables)

Most Americans will say either "groshery" or "grosery" (eliminating the "sh" sound for just an "S" ) Hope this helps !

That word can be pronounced as three syllables, or as two. If pronounced as two, then the C, theoretically pronounced as S in that word, may assimilate to the following R, thus being pronounced as Sh. In my opinion, the three-syllable pronunciation is overly careful, while pronouncing the S as Sh is a bit regional.

in the UK we would pronounce it Grow-sir-ree in 3 syllables. Usually we would use the word "shopping" not groceries so you could avoid it!!

either are correct in America. as others pointed out, when said at normal speed it sounds like grow-sree. best of luck