Congratulations to your daughter! You (she) will need to have iTunes on the computer with the music library she wants to sync her iPhone with. The first time you open the program after installation, it should ask if you want it to search for all the music in the computer that you want it to import. Once you have that done, just connect her iPhone and the program should automatically detect; it will check if the phone is on the latest firmware, and you can choose whether or not you want to download it and update now, or later (just choose Cancel). Syncing should take place immediately afterwards.
If the music she/you want on the phone isn't being synced, after the phone is connected and detected, go to the Music tab and select what you want synced on the device (ie, entire music library, selected Playlists..). You can add more music or movies to your iTunes by going to the File -> Import and choose if you want to import by individual file or by entire folder.
Congrats again, and let us know if you have any problems. I think you/she should find the process fairly problem-free.
