What I would suggest:
When you have the new iMac (this assumes there is nothing on the new one that you want):
Boot to INTERNET recovery, then
erase the internal drive.
Next, install a completely new, fresh and clean version of the OS onto it.
When done, you should see the setup screen.
BUT
STOP RIGHT HERE for the time being. Don't touch anything more yet.
Just let it "sit there" for the time being, because you have other preparations to make.
Now, you need to go to the OLD iMac and prepare a proper, cloned, "migratory backup".
Download CarbonCopyCloner from here (you can do this right now):
Carbon Copy Cloner - Download
Use CCC to create a cloned backup of your OLD iMac.
Now you have an up-to-date copy of your data "at the moment you migrated".
Next, connect the cloned backup to the NEW iMac.
Begin "clicking through" the setup procedure.
At the appropriate moment, setup assistant will ask if you have data to migrate from another Mac or drive.
YES, you do.
"Aim" setup assistant at the external backup, then give it some time to "digest" everything (you don't get much feedback while this is happening -- BE PATIENT).
Setup assistant will present you with a checklist of stuff to migrate. I suggest you just accept everything and "let 'er go".
Again, it will take some time.
When done, you should see your login screen, and after you log in, it should look pretty much as it did on the old iMac.
THERE'S ANOTHER WAY YOU COULD DO IT:
You didn't tell us which version of the OS you're using on the old iMac, and what year the new iMac is.
IF the version of the OS you're using RIGHT NOW can boot the new iMac, you could do it this way:
a. Download CCC from the link above
b. Use CCC to create a bootable cloned backup of the OLD iMac on an external drive.
c. Boot the NEW iMac with the cloned backup
d. Use Disk Utility to ERASE the internal drive on the NEW iMac
e. Use CCC to "clone over" the cloned backup to the NEW iMac.
You want to do a "test boot" from the external drive BEFORE you do this, to be sure that the cloned backup will actually boot the new iMac !