at the bank - the bank is a point in space, like on a map. It doesn't matter if you are standing on the street outside or actually inside the building.
in the bank - the bank is a room, a space with three dimensions, and I am enclosed by it. I am not on the street outside
inside the bank - not wrong but unusual in most contexts. "Inside" is like "in" but with emphasis. Why do we need to emphasise "inside the bank"? Consider this example headline: we spoke to a man who had spent three years inside the secret world of the [name of criminal gang]
e.g. I put the key inside the box and locked the box, to keep the key as safe as possible.
Consider these two situations: a) "Hello, come in! It's nice to see you! Let me take your coat."
b) "Oh, the rain is terrible today. Come inside quickly and dry yourself off. Don't stay outside for a moment longer."
Note that "inside" can be used as an adverb (like in the last example) and as a noun ("the inside of... ")