• to go shopping
We went shopping for a gift at the mall. 我们去购物中心买件礼物。
Before you go shopping, think about if there is anything you really need.
• to do the shopping (侧重指去买的是我们生活中的必需品)
Few women really enjoy doing the shopping.
•to shop (in store or online)（shop做动词）
We don't usually shop at Walmart. 我们一般不去沃尔玛买东西。
• to do [a bit of, a lot of, some] shopping
• to do shopping 不是固定表达，但在句子可能出现。
I normally do shopping on Saturdays and gardening on Sundays.
注：澳洲人懒，经常把 to do the shopping 中的the省略。