You can save the contacts to an SD card and then transfer the card to the new phone. You can save the contacts to a file and transfer it to a computer via USB cord connection and then copy them back to the new device the same way.
You can also save the contacts and other files to a cloud or Google account and then (before deleting them off the older phone) download them from the account to the new phone.
I suggest not deleting them until you know for certain that they are now on the new device because errors happen and you don't want to lose all the info.