Going from windows itunes to Mac itunes casues some issues....trust me I tried.
I think the easiest thing in the world for you to do is to turn on her phone, back it up to icloud, actually start a backup by turning wifi on and going into icloud backup and click backup now...then when its done, I'd just plug the phone in the new computer sync it so it marries to the Mac itunes, then I'd open icloud and backup from the cloud so everything is there.
If anything, backup contacts and music separately, but icloud seems to be a good idea for you at the moment.
Just remember, backing up to icloud will not backup music...so you will have to backup the music another way....icloud backups are the settings/apps, etc on the phone, including contacts, calenders, and everything....which I find the most important....music I have backuped up on 3 different hard drives, so I will never lose it..