go to (the) market

I daresay you are correct about the AmE BrE difference.

Also, there is a difference in BrE between "going to market" and going to a (or the) market".

A farmer may talk about going "to market" generically to buy or sell animals or produce, so in BrE, you might say, "I go to market with my mother every Sunday, but nobody wants to buy her!"

To market, to market, to buy a fat pig
Home again, home again, jiggety jig

To market, to market, to buy a fat hog
Home again, home again, jiggety jog

(Rhyme sung or chanted by adult while giveing small child a "ride" on his or her knee)

"The market" means, in BrE, a specific market. "The market which is held every Sunday in the Town Square in Cambridge"

See also the BrE "In hospital" versus the AmE "In the hospital"