Q: My music isn't syncing to my new computer?

do you have your old computer with your music on it??

Even if you manage to re-download all your files (this will only happen with iTunes Store purchases) you will still end up having to erase your i-device because it is not syncing to the same library.

Solution? Read on...

Moving a library: Quick answer if you use iTunes' default preferences settings:  Copy the entire iTunes folder (and in doing so all its sub-folders and files) intact to the other drive.  Hold down the option (alt) key (shift on Windows) and open iTunes.  At the select prompt choose the iTunes Library.itl file in the copied iTunes folder.

When this is to a new computer put the copied iTunes folder in the default location of Macintosh HD > Users > *User Name* > Music  (or Home > Music) then you don't even need to start with the option key held down, iTunes will automatically look for it there.  (Make sure there isn't anything already in the iTunes folder there that you want to keep since you will be replacing it with the one you are moving.)  Also remember to authorize the computer in iTunes to your Apple ID if you use this to sync to Apple mobile devices.

Consider using Migration Asist to move all your files from the old computer to the new.

If you do not have your original computer or a backup then you can only re-download those your purchased from Apple (and only if that is possible where you live).  You can try to recover files from the i-device but you will have to buy special software to do it and you still won't get your library as it was.

Your i-device was not designed for unique storage of your media. It is not a backup device and media transfer was planned with you maintaining a master copy of your media on a computer which is itself independently backed up against loss.  To use a device with a different setup you transfer the old library from a computer or a backup directly to the new setup, not the device to the library. Media syncing is one way, computer to device, updating the device content to the content on the computer, not updating or restoring content on a computer. The exception is iTunes Store purchases which can be transferred to a computer.

Redownload or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer - https://support.apple.com/en-us/HT201267 - "This feature works only for content bought from the iTunes Store."

For transferring other items from an i-device to a computer you will have to use third party commercial software.  See this document by turingtest2: Recovering your iTunes library from your iPod or iOS device - https://discussions.apple.com/docs/DOC-3991  Even this method may not fully recover what you had in the library originally. For example if in order to save space when syncing you had converted music files to a lower bitrate, or photos to a lower resolution,  it is those lower quality files you will recover.