The idea is to copy the media files and a library backup to the new computer. Then restore the library backup on the new computer and point the new library to the media files.
If you have both of the computers on a network and have them set up to copy files between them, then you can copy the files for the old to the new system over the network. You can just use windows explorer to do that
Otherwise, you can copy the files to the external drive, connect the external drive to the new computer and copy the files to the new computer. You can just use windows explorer to do that. Using an external drive may be faster depending on the network.
You should be able to do a backup of MC21 (File - Library - Backup), move that backup to the new system, and restore the backup to the new system. I think a MC21 backup can be restored to a MC24 but am not sure. Let's hope so.
If the file structure on the new computer is the same as the old computer, then the library should just work. That is, if the music files are in the same directories, like c:\jriver music\, then it should be OK. If the directory structure is different you can use the Library Tools - Rename, Move, Copy function to reset the directory structure. The easiest way is to just reproduce the file structure for the media files on the new system.
The Moving Files Wiki Entry has more details, including a link to a very detailed description of the process.
If that makes sense, give it a try. If not, ask some questions.