Cleaning up my eMac

Yep, re-installing from the original disks will do the job nicely as long as you choose FORMAT and INSTALL from the options.
That will wipe the drive and put down all that was there when you first took it out of the box.

You did not mention what OS you were using on that eMac.
If it is OS X, then all you need to copy, to save all your stuff, is the HOME folder. That contains all your documents, pictures, tunes, movies, etc.
If it is System 9.x, then copying the entire drive is a good bet as there is really no logical place where all that stuff would be stored.

However, if you can wait until you get your new iMac before giving the eMac away, I have another, easier, suggestion.
Purchase a Firewire 400 to Firewire 800 cable. You should be able to find one at your local Best Buy or someplace like that. If you have an Apple store close by, they have them too.
Fire up you new iMac, follow the instructions, and when it gets to the point where it asks if you have data that you would like to transfer from another Mac, say Yes.
Follow the instructions, connect the Firewire cable to each machine, and all your data will be magically moved from one machine to the other and, wait for it, be put in the correct place. All your network settings and other preferences will be set too.
The wonders of the Migration Assistant.

P