All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
First of all, we thank you –dear questioner –for your interest in learning the Arabic language, which is the language of the Quran. Arabic is a means for understanding the texts of revelation from the Quran and the Sunnah. Therefore, every Muslim must learn it according to his ability so that he would learn the matters of his religion.
The Kuwaiti Fiqh Encyclopedia reads quoting Imam Ash-Shafi'i: “Every Muslim must endeavor his utmost in learning the Arabic language so that he can use to utter his testimony that La illaaha illa Allah (None has the right to be worshipped but Allah) and that Muhammad is His slave and Messenger, and to be able to recite with it the Book of Allah, and utter the Thikr (mention of Allah) that is ordained on him such as the Takbeer (saying Allahu Akbar during the prayer), utter Tasbeeh (saying Subhaana Allah –praise be to Allah) which he was ordered to utter, utter Tashahhud, and so forth.” [End of quote]
As regards the method of learning, then it should be with every way possible; the most useful of which is to join a school or Center, or through the Internet if possible.
Fearing to make a mistake when reciting of the Quran should not prevent the person from reciting it if one exerts all efforts in learning the Quran. ‘Aa’ishah narrated that the Prophet said: “The one who is perfect in the recitation of the Quran will be with the honorable and obedient scribes (i.e., the angels) and he who recites it and finds it difficult to recite and does his best to recite it, will have a double reward.” [Muslim]
You can listen to a professional reciter the part of the Quran that you want to recite, then recite it afterwards so that your recitation will be in conformity with a correct recitation. The recitation should be learnt by listening to a Shaykh reciting the Quran.
This is in relation to reciting a Quranic text. As regards the reason of the difference of marks [such as Dhammah, Kasrah, Fathah] on the last letter of the same word such as:
باسم ربك . تبارك اسم ربك . سبح اسم ربك الأعلى
[in the Name of Your Lord; Blessed is the Name of your Lord; Glorify the Name of your Lord]; then it is due to the difference of factors affecting the grammatical arrangement of words (according to whether it is the subject, object, etc. in a sentence); so some require Dhammah, or Kasrah, or Fathah [on the last letter of the word].
Allah knows best.