I like [go] shopping.

I like to go shopping.
I like going shopping.

这句话有两种表达方式：I like go shopping. 这里的“go shopping"是作为一个名词“购物”。
也可以说：I like going shopping. 这里把“go" 看成单独的动词，表示一种习惯所以用like doing sth.

I like going shopping

I like shopping 才对

going

going