There are a great many methods and tips on how best to teach your child about music. Of course, all children are different: what is easily given to one can cause difficulties for another, therefore there are no general, universal methods here. Nevertheless, it would be useful for all parents to adhere to some rules that will help the child to familiarize with music.
Even when the child is very small, try to turn on music while he is awake - naturally, quiet, melodic, best of all in minor tones. Sing songs to him yourself. Of course, refrain from overly harsh, rhythmic sounds! A heavy metal style of music is completely out of place here, even if you are an ardent fan of it.
When he's a little older, start teaching him rhythm. Clap it with your palms to the beat of the music, swing it, sitting on your knee. After he starts walking, teach him to stomp his feet to the beat of the melody. Learning, as a rule, goes pretty quickly and gives the kid genuine pleasure.
Teach your toddler to distinguish between sounds by pitch. At this point, the parental fantasy can roam wonderfully! Simulate any sounds, such as the buzzing of a mosquito or bee, the hum of a cow, or the whistle of a steamer siren. And be sure to indicate: "This sound is high!", "And this one is low!"
After that, you can already move on to memorizing how this or that note sounds. It is best if you have a piano at home. Having chosen to begin with any two keys corresponding to very different sounding notes, several times alternately press each of them. Make sure that the child has memorized both the sound and the name of the notes. Then ask him to turn away, or cover his eyes with your palm, and press one key. The child's task is to accurately guess which note has sounded. Gradually complicate this exercise: increase the number of keys, reduce the intervals between them.
Most importantly, be sure to include elements of the game in the training! The kid should take it from the very beginning not as a tedious boring duty, but as fun entertainment. Especially when he begins to learn how to independently extract sounds. Moreover, here you can use not only musical instruments (including toy ones), but literally "whatever comes to hand." For example, bottles filled with water to different levels, bells, metal tubes.
Whether to send a child to a music school is a personal matter for parents. In any case, you should take into account the desire of the child himself, to find a good teacher.