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Want to bakcup and transfer android data to computer? You should use a 3rd party tool, Samsung Messages Backup is the one highly recommended for you. which is a powerful utility that specially designed for android Mobile users to transfer and backup your messages and contacts between mobile phone and desktop computer.
iPubsoft Android Desktop Manager is a trustworthy software designed to easily transfer contacts from Android to on computer directly. All contacts information will be merged perfectly. Moreover, it's not only a simple Android Contacts to PC Transfer software, but also an amazingly Android printer.
If you had the previous device linked to a Google Account (which you say you did), your contacts should automatically synched. There should be no need to import anything. Log onto Gmail on a laptop/desktop and on the left side under the big "Google" you should see "Gmail" with a drop down menu. When you click "Gmail" it shows "Gmail", "Contacts", and "Tasks". Click Contacts and it should show all of your contacts with their info. If it doesn't, then they aren't backed up, although I don't know how that would happen. If they are there and your phone is linked to your account, they should be on that phone.
If everything looks good, go into your phone settings and find "accounts and synch." Make sure you have auto-synch checked.