If your Contacts are associated to your Gmail account, all you have to do is set up the account in Settings - Mails, Contacts, Calendars and enable Contacts there. If they are not, use an app like "Copy my Data" to get them to your new device. The app I mentioned will save everything to Google Drive, and you'll be able to add them to your new iDevice afterwards. Copy my Data will also transfer photos and videos from your SGS3 to the iPhone.
Music has to be transfered to a computer, added to iTunes and synced to your iPhone, afaik.