As a native speaker, I wouldn't say "do the shopping." Say "I will go shopping" if you're talking about the future. (I will go shopping in two days.) Say "I am going shopping" if you're about to go or are going very soon. (I am going shopping in an hour.)

Thank you so much

"I will go shopping" is correct. You can also say "I will do the shopping".

Go shopping is correct. In this sentence "shopping" is an activity. "Go to" would have to be followed by a location (I will go to the store. I will go to church.) There is one exception to this rule, though - we don't say "go to home" even though home is a place.