If you have saved contacts to your Google account, which I highly recommend, the contacts will automatically download to the new phone once you add your Google account to it.
Since you have not saved contacts to Google you need to export/import them to Google. The steps below will export contacts from the old phone and import them to Google for backup and transfer to your new phone.
This is the basic steps to export contacts as a .vcf file, import them to Google (aka Gmail) contacts and then send them to another device. The exact steps may be somewhat different for your device. Consult your user manual for details.
1. Open the Contacts app and tap Menu.
2. Select Import/export, then Export to storage.
3. A box will open with the location the .vcf file will be exported to. Write it down. Proceed with exporting contacts.
4. Transfer the .vcf file to a computer or cloud storage for upload to Google. Or, upload can be done directly from the phone.
5. Log into your Google account and open Contacts. If you are uploading directly from the phone, you will need the desktop version of the Contacts webpage in your phone browser. You cannot use the mobile version.
6. Click on More. In the drop-down list select Import.
7. In the next box select Browse. Navigate to the exported .vcf file and select it. Click on Import.
8. Once your Google account is added to the desired device, sync contacts and you're done.
Android since v1.0. Linux user since 2001.