My old computer is not being detected by my new computer... it is looking for other sources. What do I need to have turned on in Sharing on my old computer?

As already stated, the best way is to create a Time Machine backup, then restore that backup following these steps: Setup Assistant and Migration Assistant - Apple Community.

For Time Machine, you need an external hard drive. Apple sells external hard drives, such as the LaCie Rugged USB-C + USB 3.0 Portable Hard Drive - Apple. Regardless of which drive you choose, it needs to be formatted. Follow these steps:

1. Go to Disk Utility (Finder > Go > Utilities)
2. In Disk Utility, select “View” > “Show All Devices”.
3. Select the top and non-indented drive in the external section of Disk Utility that matches the name of your external drive. 
4. Click Erase
5. Name: any name you want
6. Format: Use Mac OS Extended (Journaled) if you are using macOS Catalina or earlier, or APFS if you are using macOS Big Sur.
7. Scheme: GUID Partition Map
8. Click Erase

Finally, setup Time Machine: Back up your Mac with Time Machine - Apple Support.

Jack

See if you can connect the two Macs in Target Disk Mode with the old Mac in the Target Disk Mode. Then you should be able to access the files to bring them over to your new Mac.

Please connect the old computers Time Machine External Hard Disk to the new computer and run Migration Assistant . It will find the Time Machine drive. Never use WiFi to migrate it will fail almost every time!!!

Very simple, you buy one! Operating ANY computer without a backup is shortsighted and very foolish. The reason you backup is becuase you will lose data and sometimes extremely important and irreplaceable data such as photos, music, movies, finacial information, critical paperwork etc. Time Machine ensures your computer is backed up regularly and safeguards your infomraiton.

What if I don’t use or have an external hard disk?