I am told that I cannot use two "ing" verbs together, as in "I am going shopping." Is this correct?

You are right, but 'shopping' in this case is not just verb+ing, it is a special verb called "participle" which is defined as a verb that functions as a noun/adjective. Thus, in I'm going shopping, there are not used two "ing" verbs.

A simple test is that a participle can be replaced by an infinitive (generally to+verb), without changing the sentence meaning. So, you can say:

I'm going to shop.

I am going shopping is OK, I'd say rather I'm going to do some shopping 

this is correct or i am going to do some shopping

Not quite so. These are called collocations. Your sentence is fine as it is. The "ing" form stays the same no matter what tense you use:

- He has been going shopping/skiing/hiking.

- She is going shopping/skiing/hiking.

However, with some verbs usually followed by either the gerund (or the ing form if you like) or the to infinitive such as "begin" and "start" we don't normally use two "ing" verbs :

- It's starting to rain. (not It's starting raining)

 I agree with experts answer. Thanks for the invitation. .

In spoken English it is perfectly correct but in grammatical point of view, change would be applied as I am going to shop some items or I am going to do some shopping today and so on. 

In spoken English it is  OK but grammatically it is not accepted to be correct. It will be I am going to do shopping.