To do the shopping- You are going out with the intent of buying particular goods. To go shopping- You are going out for a browse around the shops and have not yet decided what you are going to buy.

Toi muon mua cai nao (Intend to buy someting ) =Do the shopping Toi di mua cho nao ( Go out to buy somewhere ) = Go shopping

"DO THE SHOPPING" the person whose doing to buy the goods.. "GO SHOPPING" is the one who will travel to shopping mall.. But they have the same sense.