A : After having lunch, where do you want to go today?
This piece of the conversation suggests they have met for lunch; that it was planned that after lunch, A would then show B around the city/take him sightseeing...something like that. And this piece of conversation is A asking B what he wants to do.
If A and B met for lunch, and are casual about what they then do e.g. shop at Harvey Nicks or go to a movie, A would say:
"What do you feel like doing after lunch/(or - if said early in the conversation, the appetizer stage,) later?"
If they had been sightseeing all morning, and so had some definite places in mind, then the kind of expressions used would be:
"What do you want to see/where do you want to go/where shall we go/what shall we see/what do you feel like seeing/doing (after lunch)?"
B would not reply using the word 'care'. He would say, "I don't mind. What do you want to do/see", or more likely, "I'm not fussy. What..."