Q: Transfer music to new IOS / ITunes install via iPhone?

Not sure what you mean.

Do you not store everything on your computer and/or cloud service?

You should. Nothing should be stored on your iPhone alone.

The music sync is one way - computer to iPhone.   The exception is iTunes purchases:  File>Device>Transfer Purchases

You should be importing your pics to your computer regularly as you would form any digital camera.

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 from a device. The exceptions are iTunes Store media purchases which can be transferred to a computer.

Re-download or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer - https://support.apple.com/HT201267 - "This feature works only for content bought from the iTunes Store." As of iOS9 it no longer does apps which now must be re-downloaded directly from the 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.

If you subscribe to Apple Music, tracks which are not part of content you have purchased or uploaded cannot be transferred and have to be downloaded directly from iCloud.

For other items see:  Import photos and videos from your iPhone, iPad, or iPod touch - https://support.apple.com/HT201302 - these are ones you have taken yourself.

For non iTunes Store media you are totally responsible for safekeeping it. A backup of a device does not transfer media because it is assumed it all originated from the iTunes Store or the computer in the first place and is safely kept there, or on a backup). We do not know why it is designed this way but I would hazard to guess it is at the behest of the media companies who get to tell Apple if Apple may sell their media, and who are worried about piracy.

iTunes: About iOS backups - https://support.apple.com/HT204136 - including a list of what is backed up.

If your Mac is not Compatible with Sierra but you Have Not Upgraded to El Capitan  - https://discussions.apple.com/docs/DOC-10692