It looks as if Person B is saying that he might go shopping, but that he is not certain. It is common in very informal communication to append a question mark to indicate uncertainty. In speech, this would usually be inflected as a question.

It's a sort of question, but not directed specifically at A; B is rhetorically playing out a question to himself. "Shall I go shopping?"

It might also invite input from A: "I was considering going shopping ... unless you have a better suggestion?"

It's an open door, not a real question.

Person B is answering person A's question by a ‘suggestion’. If person A has something to answer to that (“I can't stand shopping, why wouldn't we go …”, which is often what he has in the back of his mind when he asks for B's plans), then the question mark is welcoming him to express it.

If person A has nothing to answer, then it's not a question for him, just a sign that person B is unsure about these plans.

_{At least, that's the way I use that, and I do that very often.}

B is putting forward a possibility in front of A which he himself is not sure of and sort of expects A to give him an answer which may or may not be a definite one.

Question mark in such a scenario would also indicate that B has a certain faith in A's judgement and that he expects that his answer will not be a futile one.

I interpret this as B's question to himself, not to A, meaning "maybe I should consider going shopping?" And it's probably not even a real question, although the vocal pattern is the same as for a question; when written, a question mark at the end is the only way to indicate that intonation.

It could mean B is not sure he would go shopping.
If the question were "What should we do today?" or "What are we going to do today?" I would take "Go shopping?" as a suggestion for going shopping.

I use a question mark all the time, to me it means all of the above comments and to emphasize the point I'm trying to make and/or hopefully the other person will give it more thought.

It could be an Invitation to person A to accompany person B going shopping.

Update for More Information:

Well, say for instance I wake up, and someone I normally hang out with asks me what I am doing today. It's not uncommon for this to infer that person A wants to do something together. They could say, What are we going to do today?. However, Person A asking me what I am going to do today could mean they want to know if my calendar is clear, possibly to invite me somewhere or to do something.