The difference is in which shop you are going to.
I am going to a shop means that you are going shopping, but you haven't said which shop you are going to.
I am going to the shop means that you expect the person you are talking to to know which shop you are going to. Maybe there is only one shop in your area, or you have already decided which shop to go to, so they know which one you mean.
Both of these sentences mean that you will only go to one shop, but "I am going shopping" does not specify this. If you said "I am going to the shop to do some shopping" then it's clear that you are going to one shop. If you did not say how many shops you are going to you could be going to every shop in town and "I am going shopping" would still be correct.