If you have replaced your old iPhone with a new one, you can easily transfer all personal data. iPhone 5 users can do that with iCloud, users of older models must transfer data using iTunes on a computer.
Steps
Method 1 of 2: Switch iPhone using iCloud

Step 1. First, take the old iPhone you want to transfer data from

Step #2. Tap on Settings

Step #3. Tap on “iCloud
”

Step #4. Tap Backup

Step 5. Now check if the button next to "iCloud Backup" is moved to the right, and if so, check if the last copy was made today
- Slide the knob to the right if necessary. Your device will now copy all personal data on your iPhone to iCloud.
- If you haven't set up iCloud yet, you can sign in first by tapping "Settings" and then "iCloud".
- You need to activate iCloud with your Apple ID and password.

Step 6. Put your old iPhone away and get your new iPhone

Step 7. Turn on the iPhone and go through the initial settings
You must first provide certain information, such as your language and country, your Wi-Fi network and more.

Step 8. Select "Restore iCloud Backup"

Step 9. Tap "Next" and enter your Apple ID and password

Step 10. Select the time and date of the most recent copy and tap "Restore"
Wait for the device to restore the backup.

Step 11. Wait for your iPhone to reboot after the copy is restored
After that you can start using your phone.
Method 2 of 2: Switch iPhone using iTunes

Step 1. First, take the old iPhone you want to transfer data from

Step 2. Connect iPhone to computer with iTunes installed

Step 3. Click "File" (Windows) or "Archive" (Apple)

Step 4. Click "Devices" and select "Back Up"
Now iTunes will make a copy of the data on your old iPhone.

Step 5. Wait for iTunes to finish making a copy
The process may take a few minutes depending on the amount of data on your iPhone.

Step 6. Click "iTunes" in the menu bar, then click "Preferences"

Step 7. Click on the "Devices" tab

Step 8. Check if the copy was successful by looking at the date and time of the last copy

Step 9. Disconnect your iPhone from your computer

Step 10. Turn off your iPhone
Remove the SIM card from the old iPhone and insert the SIM card into the new iPhone

Step 11. Turn on the new iPhone

Step 12. Go through the initial settings when you turn on your iPhone for the first time
You must first provide certain information, such as your language and country, your Wi-Fi network and more.

Step 13. Select "Restore iTunes Backup"

Step 14. Connect your new iPhone to the same computer you just used to make the copy of your old iPhone
iTunes will recognize the new device and will display "Welcome to your new iPhone".

Step 15. Select "Restore from Backup" and select the time and date of the most recent copy

Step 16. Click on "Continue"
Now iTunes will start transferring the data to your new iPhone.

Step 17. Wait for your iPhone to reboot after the copy is restored
After that you can start using your phone.