You didn't mention what neighborhood you live in, but if you are on the northside, Old School of Music on Lincoln Ave. in Lincoln Square or Armitage in Lincoln Park is probably one of the best music programs you will find anywhere. Classes in every instrument imaginable, even ones you didn't know existed, including classes geared towards toddlers or teens or any age in between. And they have always been very reasonable and offer scholarships for those in need.
On the south side, I know the South Shore Cultural Center at the end (beginning) of Lake Shore Drive has a variety of programs, many of them free.
This is in addition to your local park and/or library, both of which always offer programs throughout the year in Chicago.
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