Using a Google Account:

Using your Google account you can easily transfer the contacts stored on the SIII / S4 Mini to the SGS5 Mini by first syncing them with a Google account. Once the first step is complete, you can then sync the contacts from the account with your Galaxy S5, and you're done. The steps are explained below:

1. On the SIII / S4 Mini, register your Google account
2. From the home screen, press "Menu"
3. Tap "Settings" > "my accounts"
4. Go to "Google" and then tap your "email address" or "user name"
5. Tap "Sync now". Alternatively, clear and mark the check box next to "Sync Contacts"
6. Switch over to the Galaxy S5 Mini and tap "Apps" while on the home screen
7. Go to "Settings"
8. Scroll to "User and Backup" and tap "Accounts"
9. Choose an account and press "menu"
10. Tap "Sync all" and the contacts should be available on the S5 Mini

Transferring Contacts via "Import / Export":

The "Contacts" application pre-loaded on all the Android Samsung smartphones oftentimes comes with an "Import / Export" function which can be used to export the contacts on the SGSIII Mini / S4 Mini in the form of a vCard file, which can then be imported by the Galaxy S5 Mini

1. Launch the "Contacts" app on the older Mini
2. Go to the "Contacts" tab
3. Press "Menu" > "Contacts to display" > "All contacts"
4. Press "Menu" and tap "import / Export"
5. Tap "Export to SD card"
6. Write down the VCF file's name for future use and tap "OK"
7. Launch the email client on your phone and compose a new email
8. Attach the VCF file to the email and send it to your own Gmail address
9. Switch over to the Galaxy S5 Mini and then open the email
10. Download the attached file and launch the "Contacts" app
11. From there, press "Menu" > "Import / Export" > "import from SD card"
12. Select an account if necessary, and load the desired vCard file

NOTE: you can transfer the VCF file from the Galaxy SIII Mini / S4 Mini to the Galaxy S5 Mini by connecting your smartphones to a computer via USB. Then, manually copy the vCard file from the older Mini smartphone to the Galaxy S5 Mini's "sdcard" directory.

Using "Send Anywhere" to Transfer Contacts:

Finally, you can also use third party applications to get your contacts transferred from one phone to the other. A popular and useful third party app is "Send Anywhere". The application allows you to transfer various types of files and contacts via a P2P connection, so you are not required to have access to a computer.

1. Go to the Play Store and download "Send Anywhere" on both of your Mini devices
2. After the application has been installed, open it on both devices
3. On the Galaxy SIII / S4 Mini, tap "Send" > "Contacts"
4. Select the contacts that you wish to share and then tap "Send"
5. Finally, select the desired transfer method. By tapping "Receive" on the Samsung Galaxy S5 Mini, you can enter the randomly generated transfer code from the SGSIII / S4 Mini. Alternatively you can also tap "Generate QR Code" on the older Mini, and then scan the code with the Galaxy S5 Mini's camera to begin the transfer