I agree with MJS: to do the shopping means to engage in a regular household activity, and involves purchasing food and household supplies.
To go shopping means to go somewhere to buy something of any kind.
Few women really enjoy doing the shopping (that is, going to a grocery store to buy bread, or meat, or milk), but many women love to go shopping (that is, going to stores that carry clothes, or shoes, or accessories.)
Ariel66, you've described the difference very well already, needing only a couple of minor refinements.
"Going shopping," which is exactly as you describe, is a pastime or hobby for some people. It generally does not involve food.
Note that "going shopping for [something]" is different: It means that you are specifically looking for one or more items, which could include food if it was a special occasion or unusual type of food. "I'm going shopping for milk" would sound weird, but "I'm going shopping for the ingredients for tonight's dinner party" is normal. "Going shopping" usually implies visiting more than one store, although one could certainly "go shopping" at only one store if it were a big store with a large variety of products.
"The shopping" is any routine shopping for household necessities. It does not have to be food only, as an ordinary weekly trip to the supermarket is likely to include many non-food items, such as paper towels, toilet paper, cleaning supplies, etc. By extension, people sometimes say the "the shopping" in non-domestic settings. For example, an employee might "do the shopping" for his office at an office supply store, meaning that he's going to buy the daily necessities of the office: copier paper, Post-It notes, pens, etc. By further extension, we might well say that he was "doing the shopping" for the office even if he was - as it most likely - placing the order on line with an office-supply company.