How to sync iTunes on a new computer without losing current content

Question

I just bought a new Macbook pro with retina display. I have an iPhone 5C and have a lot of songs on it, I want to sync new music from my new macbook onto my phone but don't want to lose the music already on my phone.

Answer 1

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 new computer you transfer the content from the old computer (or its backup) directly to the new computer, not the device to the new computer. 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.
iTunes Store: Transferring purchases from your iOS device or iPod to a computer - http://support.apple.com/kb/HT1848 - only purchases from 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
If you only have iTunes Store purchases transfer them to the computer, then erase sync them back again.  Either that or transfer the iTunes library to which the phone is actually synced.  Quick answer if you use iTunes' default preferences settings:  Copy the entire iTunes folder (and in doing so all its subfolders and files) intact to the other drive.  Open iTunes and immediately hold down the option (alt) key (shift on Windows) so you get a prompt to select a library, then guide it to the 'iTunes Library.itl' file in the moved iTunes folder.
If this is to a new computer and you put the copied iTunes folder in the default location of Macintosh HD > Users > *User Name* > 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.)