There are basically two parts to your iTunes library - the media files themselves, and a handful of additional data files that tell iTunes where everything is located.
On the OLD Mac - launch iTunes. Click the iTunes menu at the top, then choose Preferences. Click the Advanced tab. This will show you the location of the iTunes folder ON the machine that contains all of the data files. You need a complete backup of this folder. Get yourself a different external drive and copy this folder to it.
On the NEW Mac - launch iTunes and find the media library folder like you did above. Shut down iTunes. Launch Finder and locate that iTunes folder. Rename it to iTunes.orig.
Access the external drive that you copied your iTunes folder to, grab the folder, and drag it over to where the original folder was. Let it copy.
Eject the external drive, connect the drive with all of the media files, launch iTunes. Test.
Note - with the exception of the step above to rename the iTunes folder on the new mac, do NOT change anything! Do not delete, edit, or move and files or folders - copies only.