Learn English – “I went for shopping”

For me the difference is that "for" expresses reason and "to" is for purpose. For example:

1. Reason:

   He was fired for drinking during working hours. (Drinking was the reason he got fired.)

   She was fined for driving above the speed limit. (A fine was imposed as a result of speeding.)

2. Purpose:

   You can go to the old airport to drive as fast as you want. (Driving is the purpose of going to the old airport.)

   They all come here to drink during working hours. (Drinking is the purpose of coming here.)

That is why in examples 1 and 2 we use the infinitive. My coming here is not a result of the starting or the telling. Starting a business and telling you about it are the purpose.

Sometimes the difference is not so obvious, like the fishing example above. As already commented, If I came for the fishing, it is the reason which brought me here. Whether I want to do that or watch others do it or I came to impose fines because fishing is not allowed here is not specified. If I came to fish, however, this was my purpose, so I plan to do some fishing.

Looking at the last few lines, I see that you are a bit confused about 'went' and 'been'. Well, as far as 'visiting' or 'going' to places is concerned, 'went' talks mainly about a one time event whereas 'been' talks about going and coming back.

I've just been to the bank -I went and came back from the bank.

On the other hand, if you say...

I just went to the bank- talks about your presence in the bank (probably a one time event). For instance, you tell your mom at home that if your friend comes asking you, she may reply that you 'just went' to the bank.

In the same scene, when your friend is waiting for you and you arrive at your home, you can tell him that you have just been to the bank. So, 'Just been here is -went, finished the work, came back'.

Remember the nursery rhyme? Pussy cat pussy cat where have you been? So, here, the cat had gone and came back! :)