I have a valid F1 visa (date of issue: June 14), which I can use to enter the US starting from July 24 (30 days before the start date of my study). Do I need a transit visa on July 21 to pass through the US? (I will switch planes at Chicago, then continue to Canada - I also have a Canadian tourist visa)
It seems to me that you have answered your own question: Since the F1 visa can't be used to enter the US, you do need another visa.
Some places in the world have complex rules to allow airside transit without visa or based on specific residence permits or visas from other countries but not the US. In fact, US airports are not designed for sterile transit at all, everybody goes through customs and immigration at the first point of entry and passenger flows are not separated after that (see Do I need a US visa to transit (or layover) through an American airport?).
If you are not a citizen of the US, Canada or Bermuda or eligible for the Visa Waiver Program (VWP), you need a US visa (or in some cases another US document like a green card or border crossing card) to enter the US, even if it's only to switch planes at a US airport. Even permanent residents of Canada need a visa for that (unless they come from a VWP country, obviously).
While it does involve some costs/delay/paperwork, a transit visa should however be very easy to get for you. Since you have already been approved for a F1 visa and also have a Canadian visa, the risk that you would misuse this visa seems close to nil.