If I’m on a warranty can I take my phone to a repair shop and get it repaired for free?

It depends on a number of factors. If you purchased AppleCare+, you can get accidental damage repaired for a small fee.

If it's only under the 1-year Limited Warranty, then it only covers manufacturing issues. From Apple-

The Apple Limited Warranty covers your iPhone and the Apple-branded accessories that come in the box with your product against manufacturing issues for one year from the date you bought them...If your situation isn’t covered, you’ll pay a fee. If the issue with your iPhone is ineligible for service, you might pay the full replacement value. (bold mine)

If your phone qualifies for warranty repair: Apple Repair and Repair Status Check - Official Apple Support

If you broke the device yourself, it is not covered under warranty.