If you know the shopping mall, you must say: I went to "the" shopping mall last week. If you dont know the shopping mall and that has been your first time you must say: I went to "a" shopping mall last week. If you go to the shopping mall on a regular basis, with consistency you may say: I went to shopping mall every other day or another adverbial phrase for time.

I don't think you can say the 1st sentence. The 2nd sentence is correct when you mean any mall, not a specific one, and we use "a" in that case, while the 3rd sentence refers to a "specific" mall as you used "the". So we use "a" when we don't want to specify that thing while we use "the" when we refer to a specific, particular one.

You can go to a shopping mall, the shopping mall. You can also use expletives such as: I went to the damn mall.