As for the question how characters are **taught**, one important point that all other good answers mentioning 反切 fail to elaborate, though they really give a brief & clear introduction to the traditional academic way of showing character pronunciation in dictionaries, is that when we are talking about "before the introduction of pinyin", a nationwide standard Chinese (e.g. 國語 Guoyu, 普通話 Putonghua or so) might have nor been popularized. What does this mean? Actually in the old days, as told by my 外公 Waigong, a native speaker of 贛語 Ganyu born in the 1930s, in any 私塾 Sishu, for a long period the only type of institution that served for elementary education before the introduction of western-style schools, all characters were pronounced by the 先生 Xiansheng in their dialectal versions. And kids generally recited the fundamental materials such as 《三字經》 Sanzi Jing 《百家姓》 Baijia Xing 《千字文》 Qianzi Wen by imitating their 先生 without deeper understanding, till they could fully handle the mapping between the characters and their pronunciations. In such process, no auxiliary tools for phonetic indication were used, nor were a unified one even possible, due to the huge differences between dialects.

And I shall say, from the viewpoint of a native speaker of 普通話 Putonghua, who has learned it through school education since young, 反切 could hardly have been used as a practical tool to teach young children the pronunciation of characters, since occasionally it uses more complex characters to explain simpler ones. Plus, the practice of combining syllabic characters rather than juxtaposing phoneme symbols itself shows the restriction of old Chinese scholars' understanding of the relation between syllable and phoneme, not to mention how young kids should use this imperfect tool to learn "hard" characters from scratch.

P.S. apart from 反切 which indicates a syllable in whole, from my vague memory, there might also be a **tone-indicating symbol system** called 四聲破讀記號 Sisheng Podu Jihao or something similar in ancient literatures, which puts a small circle in one of the four corners of a character, so as to indicate whether its tone is 平 Ping, 上 Shang, 去 Qu, or 入 Ru of the traditional tonal system. But unfortunately I can hardly find a reference for this. Anyone who can provide further information would be highly appreciated ^_^.