Do you like going shopping?
"Do you..." implies present tense, and "going shopping" refers to the process of shopping. So "Do you like going shopping" would mean in the general sense, inquiring about a preference: "In general, do you currently enjoy the process of shopping?".
If you were accompanying someone on a shopping trip and asked that question, it would still refer to the general sense, but it might be interpreted in the context of that trip since that trip was happening in the present, as in "Do you enjoy shopping (in general), giving consideration to your experience right now?" (The person might generally dislike shopping but could be enjoying that specific event, so it might influence their attitude.)
Do you like going shopping tomorrow?
Going shopping tomorrow is a future event. You can't enjoy it in the present because it hasn't happened yet. To inquire about someone's preference for engaging in a specific shopping event now or tomorrow, you need several changes to the sentence, such as "Would you like to go shopping now/tomorrow?"
"Would you like..." and "to go shopping" both refer to a future event ("now" is a future event in this case because you are not yet actually doing it; it means "starting very soon").
"Like" is about enjoyment. If you are really asking about desire, "want" would be a more accurate word: "Do you want to go shopping now/tomorrow?" The difference is subtle. A masochist might enjoy shopping and for that reason, not want to. For pretty much anyone else, there's no practical difference. :-)
Do you mind going shopping?
"Do you mind going shopping?" can have several meanings, with different implied timeframes. It can be a different way to phrase "Do you like going shopping?". The choice of "like" or "mind" frames the question with an implied preference, but it's asking about the same thing--a current preference.
It can also be a request. A shopping trip is needed and you're asking the other person if they would be kind enough to do it. This meaning refers to a future event rather than a current preference.