The best way to teach children the Tajweed of the Quran is through the Talqiin (imitation) method. In this method, a qualified teacher recites the Quran bit by bit, and the child imitates him and repeats after him, with the teacher correcting the child every time he makes a mistake. This way the child takes the Quran directly from the mouth of the teacher already with the Tajweed rules.
Along with this, the teacher guides the child to some of the easier rules of Tajweed such as Idhaar, Idqraam, Iqlaab, Ikhfaa, Madd al-Muttasil and Madd al-Munfassil. This helps the mind of the child get used to the art of Tajweed and its principles.
However, the scholars of Islam did not leave the children like this but wrote books that are suitable for their age and capabilities, when they were authoring books in this subject.
From the most famous books that were authored in this regard is Tuhfatul adfaal wal-qilmaan (The treat for the children and young boys) by Sheikh Sulaymaan Al-Jamzuuri. This book is lines of poetry that is easy in terms of its words and how it is put together. The author has also explained it himself in another book and called it Fathul aqfaal bi sharh matn tuhfatul adfaal (unlocking the locks by explaining the text of the treat for the children). Many scholars past and present have also written explanations for these lines of poetry.
Although Tufatul adfaal was intended for children, it is not only for them. Nowadays it is taught as an introduction text all over the Muslim world to both young and old. It is even taught at al-azhar university. This is because, alhamdulillah it contains all the basic rules that one needs in order to read the Quran correctly.