Hey lyv208,
Thank you for using the Apple Support Communities! We understand some data did not successfully move over to your new iPhone. Let's help.
When setting up your new iPhone, what steps did you take specifically? Review: Set up your iPhone, iPad, or iPod touch. Did you use Quick Start, or did you choose to manually set up your device (step 2)? Did you choose the option to restore from an iCloud Backup, Restore from Mac or PC, or Transfer Directly from iPhone (step 5)?
On your previous device, did you have Messages in iCloud turned on? Use Messages in iCloud. If so, make sure on your new iPhone, you've also turned this back on under the iCloud Settings.
For your applications, you may need to re-download those: Redownload apps and games from Apple
Here's another article that should also be helpful: About missing information after you restore your iPhone, iPad, or iPod touch with iCloud Backup
Best regards.