As a native speaker, I would use either in the example you quote.
However, there is a difference, but it is very subtle.
If I say I'm coming with you, I mean I am accompanying you to the store.
If I say I'm going with you, I mean we are both going to the store and we happen to be going together.
I know there doesn't seem to be any difference, but if we change "with you" to "without you", we can see there is.
I could easily say I am going without you, but I would never say I am coming without you.
Despite all that, use either - nobody will misunderstand you.
Hope this helps.
Student or Learner