What's the best free software to make an exact image of a hard drive and transfer it to a new hard drive?

Clonezilla. I use it all the time. It's a live boot image though. You have to download it, burn it to a CD-ROM or USB drive. and boot to it.   If you want something more user friendly, you can use Rescuezilla.  It's basically just a graphical front-end to clonezilla. Both use the same back-end technologies.  I use Clonezilla probably once a week. It's never failed me. :)  One thing to note. Whichever version of Clonezilla you use to do your backup image, you must use that same version to restore the image. They don't really tell you this in the documentation. But if you back it up using version X, and then go to restore it with version Y, you could run into issues. Not always. But it's best to just stick with the same version.

I don't know which is the best. But, I've used PING (Ping Is Not Ghost).

I don't know if its the best, but Clonezilla is a pretty good program for cloning hard drives. Its a text-based live CD and its user interface isn't particularly user-friendly though.

I've used it several times to clone a couple of hard drives without problems.

Macrium Reflect:

https://www.macrium.com/reflectfree