I was able to do a backup of the phone using iTunes, but how do I see and access the contacts?
Do I need to import them into Address Book?
Is there a better way to download contacts from iPhone to Mac?
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The best way is to use the Apple supplied sync options - either over USB/WiFi local sync to the Contacts app (Address Book on Lion and earlier) using iTunes or using iCloud.
From there, you can backup / export them as vCard or inject them into Exchange, LDAP or other services. This way the contacts are in a standard / text format as opposed to the internal storage form for iOS or OS X contacts app.
To get started in syncing Contacts between iTunes and an iOS device, get your USB cable and open iTunes. From there - type sync contacts into the help menu search field and iTunes help will walk you through the setup. With a few clicks, you should be at a window like this and can turn on the sync for that device.