CONNECT a Mid-2011 iMac to a new 2024 iMac

I would suggest making a clone backup of your 2011 iMac onto a portable USB hard drive or SSD. Then move the USB drive over to the new Mac and copy off whatever files you want.

If you connect the Macs with a cable, you may run into the issue that Apple has added restrictions on Target Disk Mode. If I understand the Support articles correctly,

• You would only be able to use the 2011 iMac as an external drive if you had a Thunderbolt connection. Making such a connection would require $80+ in specialized hardware (Thunderbolt 3-to-2 adapter + Thunderbolt 1/2 cable) that you would likely never use again.
• You could use a less expensive USB connection if you put the new Mac into a mode similar to Target Disk Mode, where it would be the "drive" and the old Mac would be the "computer". But that's probably not what you were hoping to do.

Transfer files between two Mac computers using target disk mode - Apple Support

The fly in the ointment: "Note: If either of the computers has macOS 11 or later installed, you must connect the two computers using a Thunderbolt cable."

Transfer files between a Mac with Apple silicon and another Mac - Apple Support

Says that you can use a "USB, USB-C, or Thunderbolt cable" – but this is when you are making the Mac which has the Apple Silicon chip appear to be the "external hard disk."

I would start by reading

How to Share Files Between Two Macs

Set up file sharing on Mac - Apple Support

more

macos transfer files over network - Google

so when the old one is sharing the dirs the stuff you wish to transfer

then the new one can just copy them across the network