Sorry but your question is really too general to answer.
I'm sure some brave person will rush in where I fear to tread, but this is a complex issue: you need to go and look at an English grammar book to find all the cases. There are lots of exceptions and funny rules and overlapping cases and context dependent cases and so on and so on. You could quite easily use all three:
My mother is on the shopping centre
= e.g. My mother is standing on top of the roof of the shopping centre
My mother is at the shopping centre
= e.g. My mother has gone shopping and I am at home
My mother is in the shopping centre
= e.g. I am standing outside the shopping centre waiting for my mother.
You have understood one thing correctly: a preposition is combined with what follows it to form a prepositional phrase, and they form a single constituent in the sentence. Generally speaking, words in English govern (ie. control or specify) the words that come after them. In linguistics, we say that English is right-branching, meaning that new syntactic elements come after (to the right of, in writing) the elements that govern them.
Note that there are exceptions, such as adjectives, which precede the nouns that govern them. English is not exclusively right-branching, but it is predominantly right-branching.
But what does this have to do with prepositions? Well, just as a preposition governs the noun phrase that comes to its right, the preposition itself is governed by something to its left. And in many cases, that thing is a verb. English is full of idiomatic combinations of verb + preposition, where the verb requires a specific preposition to follow it, and anything else is an error. To take some obvious examples cribbed from other answers:
I converse with you. [Not to/at/of you]
They rely on the bus. [Not with/to/at the bus]
These combinations are highly idiomatic, meaning that the correct choice of preposition cannot be predicted simply by knowing the general meaning of the words involved. So the people who ask about what preposition follows a certain word are asking a reasonable and intelligent question. The choice of preposition very, very often depends on what came before it.
Best Answer
With is an instrumental usage, as if English is being considered as a tool -- proficient with knives, proficient with horses, proficient with languages, especially English. Perfectly cromulent, but not often encountered, I suspect.
At is a punctual locative, locating some spot in a larger area or metaphoric space (e.g, in May, on Tuesday, at 2:34 pm) -- proficient at getting his deer every year, proficient at locating the fault in my argument, proficient at language and language games.
In is the general case with proficient (or skilled), though in can't take just any clause or phrase as its object; it has to at least be some activity that is learnable, repeatable, and worth repeating. These examples are terrible, for instance -- *proficient in going down to Joe's and bringing me a ham on rye, right now, *proficient in being late three times out of four, etc.
I'm sure there are many other constraints governing the object of in with proficient, and good luck in discovering all of them.