What is better?
I'm going shopping
I'm going to go shopping
I'm going to shop
I'm starting skating
I'm starting to go skating
I'm starting to skate
So, the context is different. Could anyone explain me what each sentence means. Thank you.
Although 'I'm going to shop.' is not completely natural. You could still use this phrase and be understood. I also don't see anything wrong with it grammatically either.
As for 'I'm starting to' vs. 'I'm starting ~ing', 'I'm starting to' is usually prefered/better. For example,
I'm starting to learn Japanese.
For more details, read infinitives and gerunds with start and begin.