Pozdravljanje[1]

Bok! Ja sam Mirjana iz Mađarske. Dolazim[2] iz Pečuha, ali živim u Budimpešti. (Ja) pričam govorim[3] mađarski, finski, engleski i malo hrvatski. Ja sam grafičarka[4] i radim kod[5] kuće, imam fleksibilno radno vrijeme. Budim se u 7 (sedam) i ujutro pijem uvijek čaj, rijetko kavu. Učim hrvatski(,) jer moja baka[6] je bila Hrvatica. Ona je umrila prije tri godine. Nadam se da jednom ja ću govoriti hrvatski bolje.[7]

1) "Pozdravljanje" is a verbal noun and, as such, it denotes an action, so we wouldn't use it in this context. Were you using English while writing this? That might have confused you, since English does allow a verbal noun here ("greeting").

2) The sentence is okay, but I would like to suggest the adverb "inače". It usually means "otherwise", depending on the context you could translate it into English as "besides", and it can also mean "usually, normally", as it would in this case. This is to avoid confusion, as we would we would (also) use the expression "dolazim (iz)" ("I come (from)") to denote the place/settlement we live in (which in your case is Budapest). I presume you were born (and raised?) in Pecs, and only later did you move to Budapest.

So the sentence would be "inače dolazim..." or "inače sam...", "normally I come from..." or "normally I am from..."; suggesting that what you mean is "by birth/origin", which you could do as well by using the adverb "rodom". The root "rod-" conveys in it the meaning of "birth" and "blood relation" (roditi: to give birth; porod: birth; roditelj: parent; rodbina: relatives; and so on).

3) "Govoriti" - "to speak"; "pričati" - "to tell". The languages we speak, and what we tell are stories (priča; plural: priče). Standard language is very strict about this, but in everyday language this is not that big of a deal; I'd say that most Croatians make that mistake. :D

The pronoun "ja" is in brackets because it's not necessary (you can tell from the verb who the subject is), but it's not a mistake to leave it either, if you wish.

4) Again, the same as above regarding the pronoun. You may leave it, of course, but if you should choose to omit it, just know that the word order would change to "grafičarka sam".

5) "Kod" - "at"; "od" - "from". It would be better saying "radim od kuće" ("I work from home").

6) This is fine, using the possessive pronoun (moja), but using the personal pronoun ("ja") in dative case (short form "mi", not "meni") would sound more natural. This type of dative is the so called possessive dative and, as the the word itself suggests, it denotes possession (in true sense, like owning an object, a thing; or else relation). You would literally translate this as "to me" into English ("grandma to me"), although this is not a construction that's natural in English. The sentence would then be: "jer mi je baka bila Hrvatica".

I googled a bit and found what appears to be a similar thing in Hungarian. The sentence is: Nekem nincs testvérem. So maybe you can compare it and understand easier.

7) Croatian is very flexible when it comes to syntax (word order), we can figure out the meaning, but it does sound unnatural. So my suggestion would be this: "Nadam se da ću jednom bolje govoriti hrvatski"; and I would definitely suggest leaving out the personal pronoun ("ja"), unlike the example under number 3, where you really can let it stay. Because in this case it would sound like you're trying to emphasize the subject in contrast to someone else. Like you're comparing yourself to someone who doesn't speak Croatian well, and you hope that <em>you</em> will one day be able to speak it better (than the other person).

***

Miriam, this is really great for an introduction, and I bet it wasn't easy! Our languages are very different, but they're grammatically complex in a somewhat similar way, so I'm sure you'll get the hang of it. Good luck with learning! :)