Do I have to back up my old iMac before I can transfer data to my new iMac?

Welcome to the Apple Support Communities.

Ideally, yes. You can then follow the steps here: Setup Assistant and Migration Assistant - Apple Community. If you don't have a backup drive, a great choice that Apple sells is the LaCie 2TB Rugged USB-C + USB 3.0 Portable Hard Drive - Apple.

Cheers,

Jack

Do you have Time Machine setup and running? If yes then you already have a backup.

If you don't then you definitely should set it up. It recommended to get an external HD that 's 2 ½ to 3 times the size of the drive(s) you're backing up.

And for a desktop iMac get one that is self powered. This is an excellent choice: OWC Mercury Elite Pro 7200 RPM Storage - excellent choice for Time Machine usage, inexpensive and self powered.

If you don't have a backup strategy enabled it's like driving without use seatbelts…………you don't need them till you do and then it's too late.

Ideally yes that assuming you want your migration to be smooth and troublefree. Please follow the steps in Setup Assistant and Migration Assistant and your migration will be trouble free, quick and complete.