Q: Getting songs off iphone5 onto new computer

Open iTunes and click sync music to your computer.

Re-download or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer - Apple Support

-Ethan

There is no sync music to your computer facility in iTunes unless you are referring to iTunes purchases.

The best way to handle this would be to copy your iTunes folder from the old computer to the new computer. Or use your backup from the old computer and restore it to the new computer. You need the top iTunes folder not the iTunes Music Folder below it. This will transfer everything including your playlists.

If you are transferring only iTunes purchased music you can connect to your computer and transfer purchases in iTunes.

If you are trying to transfer ripped or 3rd party music the easiest way is to use 3rd party software such as TouchCopy or Phoneview.