Although I've found live music to often conflict with naps and early bedtimes, there are various free musical experiences available to young children through San Diego County.
Poway Folk Circle
This is not specifically for children, but children seem to enjoy this group. It's an open group for folk musicians, not a band per se. They play in the morning at the Poway Craft Market in Old Poway Park on the first Saturday of the month. (There is a farmer's market adjacent to the park and the Poway-Midland Railroad runs through the park on weekends). They play at least 4 times per month, usually at Old Poway Park, but once a month at Round Table Pizza in RB.
Clint Perry and the Boo Hoo Crew
We saw this group play at San Diego Botanic Garden one evening. Little kids love them. They play some classics (like Itsy Bitsy Spider with some additional verses) and a lot of original songs with catchy tunes and positive or educational messages. They have a regular gig twice a month at a coffee house in San Marcos and sometimes play libraries and other venues.
Hullabaloo
Self-described as free-range organic kid-folk, this local band has many free and low-cost concerts around the county.
