How to migrate iMessage conversations and phone contacts from old pre-Quick Start iOS 10 iPhone to new iPhone?

The best and the easiest way out to transfer only messages, you need to enable Messages on iCloud on your old device. Once it is synced.

All messages are automatically available on all devices that use the same Apple ID.

If you have already set up the iPhone then the data created during this period will be lost if you use any of the processes given below

iPhone to iPhone, there are three ways,

1. Quick Transfer --> Use QuickStart to transfer data to a new iPhone, iPad or iPod ...
2. Restore from iCloud Backup --> Restore or set up your device from an iCloud backup - Apple ...
3. Restore from iTunes backup --> Restore your iPhone, iPad, or iPod touch in iTunes on PC ...

Instead, try transferring data manually. One by One

1. Messages, Media, Voice Memo, Keychain --> Turn on Messages in iCloud & iCloud Photos, Voice Memo, Keychain, and your content is automatically stored in iCloud and is available on all devices that use the same Apple ID
2. Contacts --> Contacts are not stored on an iPhone nor on SIM, they are synced to  Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other. Use the email account that is in your old iPhone.
3. Notes, Reminders, Calendars --> You may sync all these from any of the email accounts that you have on your old iPhone. they can be synced using  Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other.
4. Any Specific Files --> Upload to Cloud Drives (iCloud Drive, Google Drive, OneDrive, DropBox, Flickr, box, AmazonDrive, Adobe Creative Cloud and Use Files app on your new iPhone too.
5. Apps & Settings --> In any way they have to be redownloaded using the App Store. Even you had used any of the 3 processes mentioned earlier. Apart from the Apps following will also not get transferred.
   1. Content from the iTunes and App Stores or PDFs downloaded directly to Apple Books
   2. Content synced from iTunes, like imported MP3s or CDs, videos, books, and photos
   3. Data already stored in iCloud, like iCloud Photos, iMessages, and text (SMS) and multimedia (MMS) messages
   4. Face ID or Touch ID settings
   5. Apple Pay information and settings
   6. Apple Mail data
   7. Apple Music
6. WhatsApp --> Take a backup of chats and media from within the WhatsApp App. Use the same Phone number as that of your earlier iPhone and activate the account in the new iPhone too. That's it.
7. Shazam Account --> Access your Shazams on all your devices or online - Apple ...

Unlike some Android devices, contacts are neither stored locally on an iPhone nor in SIM. They are synced to an email account that you had Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other

If Contacts on iCloud, try this --> Restore contacts, calendars, bookmarks, and more using ...

Else add the email account that stores your contacts.

Apps don't transfer from iPhone to iPhone, only the app data along with app links to the App Store gets transferred.

Apart from the above following will also not get transferred.

• Content from the iTunes and App Stores or PDFs downloaded directly to Apple Books
• Content synced from iTunes, like imported MP3s or CDs, videos, books, and photos
• Data already stored in iCloud, like iCloud Photos, iMessages, and text (SMS) and multimedia (MMS) messages
• Face ID or Touch ID settings
• Apple Pay information and settings
• Apple Mail data
• Apple Music