Do native English speakers say ' I am going to the movies' when going to see a movie?
Yes, hundreds of millions of native speakers say this. This is the standard and correct phrase in American English. The British equivalent of this sentence is 'I am going to this cinema'.
You can also say 'I'm going to see a film' ( in British English) or 'I'm going to see a movie' ( in American English).
'I am going to a cinema to watch a movie' is not wrong. The grammar is correct, and the vocabulary is correct, too. Any English speaker would understand you. If you wrote this in an English exam, you wouldn't lose any marks for it. A native speaker probably wouldn't say this, however, because it is unnecessarily 'wordy'.
There is no need to give the same information twice. If you are going out to see a movie, it is very likely that this will be at a cinema, and if you are going to the cinema it is very likely that you will be seeing a movie there. Just one piece of information is enough. So you would say either:
I'm going to the movies (US)
I'm going to the cinema (GB)
or
I'm going to see a movie (US)
I'm going to see a film (GB)
Who told you that this was wrong? The person who told you this was being very unhelpful. To dismiss a correct sentence as 'total Konglish' is also rude and ignorant. Don't take any more advice from this person.