Basically, they sang different songs. And did lots of and lots of recitations.

The approach to basic literacy education before the Civil War is known as the Alphabet Method. It was a mostly oral approach where children were taught how to spell before they were allowed to read. Noah Webster published his popular speller in 1783, which included lists of words (culminating in eight-syllable words) that children were expected to learn before they could get to the reading part of reading. In effect, children were taught the parts before they were shown the whole.

They would use hornbooks and primers, usually focused on religious phrases and texts, to learn increasingly complex syllabary. The learning focused on "reading" words by each letter or syllable. For example a list of one syllable words might include "God", "grace", "peace", and "bad" ("bad" would be read by children as "bee ay dee.")

But regarding songs, they were a common tool long before the familiar one you mention as children would sing the letters to the tune of a song they already knew. A popular one was the hymn "Refreshed by Gentle Slumbers" (which sounds similar to "My Bonnie Lies Over the Ocean"). There was also a rhyme/song called "Apple Pie ABC" or even more delightfully known as "The Tragical Death of A, Apple Pye Who was Cut in Pieces and Eat by Twenty-Five Gentlemen with whom All Little People Ought to be Very well acquainted." Benjamin Harris' primer, published around 1690, contained phrases like "In Adam's fall, we sin all", "Thy life to mend, this Book attend." There was no shortage of inventive ways to get small minds to learn the nonsensical mess that is the English language.