As I understand it, there can be a slight difference. Often we use: 'go + gerund' for a recreational activity (like go hiking, go surfing etc) while we use: 'do' for chores and work (do your taxes, do the washing So 'go shopping' seems more recreational to me, like going shopping for clothes , while 'do the shopping' seems more like a chore (like doing grocery shopping).

I agree with Jessie. The phrase "do the shopping" implies that you're doing a chore. You're buying things because you need them. The phrase "go shopping" can mean the same thing ("I need to go shopping because we're out of bread"), but it can also mean that you're shopping just for fun.

"go shopping" and "do some shopping" are both great! No real difference.