Q: I just want to copy music from my phone to a new home computer

IPhone will sync with one computer at a time. Syncing to another will erase the current content and replace with content from the new computer.

You should copy everything form your old computer, or your backup copy of your old computer, to your new one.

See How to move your iTunes library to a new computer - Apple Support for instructions on copying your library from your old computer.  If you no longer have access to this, or to a backup of it, see turingtest2's user tip on Recover your iTunes library from your iPod or iOS device.  Note, though, that iDevices are neither designed nor suitable for data storage - you should always maintain an up to date backup of the iTunes library on your PC that can be used to migrate it to a new computer as well as for security against future hardware failures, virus infections, etc.