Most intriguing for non-Slavic native speaker Polish learners such as yours truly is the accusative/genitive switch when using certain masculine virile "animate" nouns
Maybe it's possible to classify it in different ways, but from what I have learnt on the Polish classes in the primary school (I am a native Polish speaker) - in both situations it's genitive (biernik).
We used, of course, the Polish cases without any problems not knowing that they exist, as we had learnt them naturally, but in primary school we had to learn them, just like different other grammar topics. And we had to learn by heard a table of cases with typical questions they answer to and with single examples of typical verbs they go with. So:
1. Mianownik - kto? co? jest
2. Dopełniacz - kogo? czego? nie ma
3. Celownik - komu? czemu? przyglądam się
4. Biernik - kogo? co? widzę
5. Narzędnik - (z) kim? (z) czym? idę
6. Miejscownik - o kim? o czym? mówię
7. Wołacz - O! (there is no typical verb and has no typical question, so here goes an exclamation instead of a question)
From my grandmother I know also, that in the past the numeration of the cases was different - the wołacz was after biernik. I don't know why they moved it to the end. Maybe because it's currently really rarely used (it's usually replaced with mianownik) and it's quite different from the other cases.
For "living" objects biernik has a different question than from "static" ones. Of course, these questions don't reflect the situation very accurately, it's a simplification. Let's look even at the error from this thread, with the dog. The question words like kto?, kogo?, komu? refer only to people. If you ask about animals, you usually use co?, czego?, czemu? (although it's still rather awkward and I would try to rephrase the whole sentence rather than use these words with respect to animals, it sounds like you were referring to an animal as to a thing). But for animals biernik looks exaclty like dopełniacz, not like mianownik.
This is still not so difficult if you realize that for some nouns biernik look like mianownik or like dopełniacz depending on the situation :)
In an e-mail to a friend you may write: "Tydzień temu wysłałem do ciebie maila...". But in an official one, you would rather write "Tydzień temu wysłałem do Pana mail...". Both of them mean: "A week ago, I sent to you an e-mail...".
By the way, we observe here an interesting phenomenon concerning borrowing words from other languages. In English, as all we know, the word "mail" refers to the traditional post. The one that goes through the Internet is "e-mail", from "electronic mail". After e-mail as a means of communication came to Poland, the Polish term for an electronic letter, after borrowing it from English, quickly got shortened from "e-mail" to "mail". Even without a change of spelling. In Polish the word "mail" hadn't existed before, so now it refers only to the electronic mail. Such changes of meaning often happen while borrowing words, but we can see here how this process looks like :)
And again by the way, anything like Past Perfect tense doesn't exist in Polish. There exist reminders of such a construction that existed in Polish in the past (200-300 years ago and before) - the "normal" past form of the "defective" verb "powinien", whose present form behaves like the past forms of "normal" verbs, and it's why its past form is "powinien był". But this is not Past Perfect, it's just a construction, used now probably only with this one single verb. Nobody, absolutely nobody, uses this construction as the Past Perfect case, replacing the word "powinien" with "szedł", "robił", "widział" etc., as it was done in the past. Let's take a sentence form my previous paragraph:
In Polish the word "mail" hadn't existed before...
Its translation to Polish is:
W języku polskim słowo "mail" nie istniało wcześniej...
which could be translated to English equally well as:
In Polish the word "mail" haven't existed before...
maybe even as:
In Polish the word "mail" didn't exist before...
although I am not sure if the last one would be considered as correct English. Maybe in the US yes, but in the UK not? I have no idea. But if we cut off this "before"/"wcześniej" and leave the beginning of the sentence up to the word "existed"/"istniało", then all three will be for sure a correct translation of the Polish phrase.
A Polish beginner in English has here the same dillema, as you learning Polish cases. One Polish expression, which can be replaced by three different ones in English. But afer some time of learning English, I am comfortable with using them. As I have already said before, I think that this comes mostly with reading texts in the foreign language.
I have also written somewhere that I am currently learning German and I have problems with German cases. Even though they are much simpler than the Polish ones. Simpler, but they work in a bit different way, and the cases that go with different words and prepositions are often different than in Polish. Not to mention that you have to remember all these endings of two types of articles and after adjectives before which goes each type of article or no article.
And I am still wondering if all these what I am writing in English is a proper English :) For example the last sentence from the pervious paragraph:
Not to mention that you have to remember all these endings of two types of articles and after adjectives before which goes each type of article or no article.
Is this "goes" on a proper position? Maybe it should be at the end? Isn't the construction of this sentence, generally, "too Polish"?