I would say:
I am knowing the town quite well.
I have been noticing that get/got is often used to make shorter the kind of sentences such as 'I have to do/make/..' to 'I got to do/make/' in which 'have' is replaced by 'got'
I am getting to know/do/make/.. might likely mean that I think I am knowing the town quite well.
Not a teacher at all