How can I transfer music from one iphone to another if it has NOT BEEN purchased from iTunes (ie., songs that were put on there from cd's I own etc.)
Only one "library" or subset of one library can exist on a single mobile  device. You can manage your non-iTunes purchased music and move it to your device through iTunes.  But you can not move it from device to device, or device back to PC.  Why not?  Doing so could promote piracy of songs (media).

You can only normally transfer music in one direction: iTunes -> Device. The reason for this is to limit music sharing, this follows the model apple created to satisfy the recording industries demands as apple demanded an end to DRM encrypted files.

Additionally, related but not part of your question, you are essentially prohibited from transferring songs from someone else's iTunes library directly to your phone.  If you attempt to do so it will warn you that you are about to replace the existing library on the device with a new one, meaning all song currently on the device will be wiped if you continue.

If the original iTunes files are not available on the computer, you can't just transfer them from the device back to iTunes. There are however apps in the app store that allow you to transfer music back to a PC, at that point you can use iTunes to put it on the new phone. I can't personally suggest any of these as I've never used them. Google the term iphone transfer music to pc to learn more and read reviews.

Another method if the original CD's are still available you could use iTunes to re-rip them to iTunes on the computer, and then move them to the new phone.

Apple also sells an add-on called iTunes Match that will allow you to download songs in your ripped library directly to your device.  However this service isn't a necessity at all, it just adds convenience over managing your library directly thru iTunes. 

This is my opinion on the subject, but if a person understands how the system works, it's not hard to follow the rules, but it does seem confusing when a person is replacing their phone and doesn't know the rules.  I would also rather have this system in place than DRM encoded music downloads from the app store.