A few of my favorite Japanese kids songs are posted here.
In my daughter's backpack each week is a summary of what the children did, learned, and enjoyed that week. There are always lyrics of two or three new songs they learned that week. Since school started, that's added up to about 35 songs so far! I thought I would learn all of them, too, by looking them up on youtube or whatever, but it's just overwhelming. All preschool teachers in Japan can play the piano, and every preschool has a piano or organ used frequently throughout each day. Songs are sung throughout the day, and the kids practice special songs for special events that parents attend, like the Autumn Festival, Halloween Party, and the upcoming Christmas Party (Tuesday). My daughter is constantly singing songs like the ones used in your Japanese course. Her favorite is "ookina kuri no ki no shita de," which she pronounces completely wrong after the first line. Very cute!