It's not really the subject that you can skip but the personal pronoun (well depending on context you can skip the subject and whatever and create incomplete sentences just like in any other language). But in Bulgarian you don't have to use the personal pronoun all the time because the conjugation of the verb shows who is doing it (first person singular you know).
Especially in this sentence: "Аз съм на 28 години и аз съм от америка." you don't have to use the second "аз" - it tells a foreigner trying to speak Bulgarian right away.
But I am sure your tutor would explain to you all of this.
Windows Vista is coming with an inbuilt Bulgarian Phonetic but I have the feeling that you are already using the Phonetic Cyrillic keyboard.
All in all, well done. When you said 300 words I could not imagine that you would be able to create 13 mostly correct sentences.