What's the difference between "Miło cię poynać" and "Miło mi cię poynać"?
Except the number of words in both sentences. What's the difference on syntactic structure and emotional expression?
~ "Miło mi Cię poznać" - "Cię" starts with a capital letter, because it's the polite way to address someone. For plural "you", it would be "Was".
You can also use: "Miło mi Pana/Panią poznać" - "It's nice to meet you, Sir/Madam."
(plural: Sirs - Panów; Madames - Panie; both genders - Państwa)
Note: Polish uses the T-V distinction.
The first phrase is mostly used in written and formal form, as personal pronouns (here it's "mi") are often omitted, because it's obvious that it is you talking; it also specifies that it is you that is glad to meet the other person.
"Miło mi " - "It is nice to me "
~ "Miło Cię poznać" - It is nice to meet you. More casual, for example can be used while being introduced to friend's friends. More common in everyday use.
Hope I explained it sufficiently, if you have more questions, feel free to ask again ;)
I just remembed, you can say "Miło mi poznać" as a kind of casual way of saying "good to know you".
Example: A: "Hi, I'm Anna", B: "Miło mi poznać" and then the converation usually moves to another topic, it's like a by-the-way introduction.
Back to your question:
"Poznać" - "to meet, to get to know" answers the question "kogo, co" - "whom, what", hence the biernik use.
There're also words similar to "Państwa": "państwo" - "a country", "państwa" - "countries".
You could say "poznać państwo/państwa" - "to get to know a country/countries" (as in to get to know the culture of a country), and it makes perfect sense. It's easy to confuse it, though ;)
I disagree, "cię" isn't any kind of polite way to address anybody (nor is "was").
It's just the basic, least formal way. It can be capitalized in semi-formal writing (when you want to be polite but still informal), for example in letters, emails, or sometimes online chats but even there it sometimes looks forced and unnatural (this depends on a person's individual perception, though and especially in less formal way of text communication it just depends on what the two people writing agree on, verbally or not).
I personally don't like too much capitalisation (anywhere) and what I especially dislike is using "polite" forms: capitalized pronouns, formal "pan/pani", capitalised formal "Pan/Pani" etc. when the overall tone of a statement is not polite at all. Like, really, don't do that when you intend to offend the other person in the other part of your message. Also, sometimes it might seem ironical, which is another reason I don't like it.
And capitalizing homonyms of the "you" pronoun even if they don't have anything to do with it, is really bad, too. (Like, "Ci" is the polite-informal "you.SING" in Dative, while "ci" meaning "these" (masculine-personal) shouldn't be capitalised).
Note: when referring to the God, you ALWAYS capitalise the pronouns (except the reflexive ones - sometimes people do it but it's, again, hyperpolite).