Upgrade from another smartphone
Moving from an Android, BlackBerry, or Windows smartphone to the iPhone 5c or 5s? Depending on how you've set your information up, it should be relatively painless to transfer it to your new device.
Mail, contacts, and calendars: If you're using a Gmail account or other POP or IMAP-based account for mail on your smartphone, it's already syncing to a central server, and you should be able to add that account to your new iPhone with few issues. Apple's iOS has automatic setup for those using Microsoft Exchange, Gmail, Yahoo, Aol, or Hotmail; you'll also be able to manually set up a POP or IMAP account for mail, LDAP or CardDAV for contacts, or CalDAV for calendars. You can alternatively use Apple's free iCloud service to set up a new email account.
Music, videos, and photos: Your new iPhone uses iTunes to sync any local music, TV, movies, and photos from your computer to the device. If you've been syncing that information with your smartphone, it's easy enough sync that data with your Apple device—you just have to know where the content is located on your smartphone and get it over to your computer. Once you've done that, add it to iTunes; to sync your photos, add them to iPhoto or Aperture (on a Mac) or place them in your Pictures folder (on a PC).
If you've purchased content through your smartphone that hasn't been copied to your computer (say, if you're using Amazon Cloud Drive), you should be able to download it to your desktop system, or, at the very least, install an app on the iPhone (like the Kindle app for book purchases) that lets you access the information.
Apps and miscellany: Unfortunately, you can't port any Windows or Android apps from your old device to your iPhone. On the upside, you may be able to find parallel versions of those apps on Apple's App Store (for instance, if you're using Dropbox on your smartphone, you can download the company's iOS app and continue to access your Dropbox data). If you have apps with valuable information you don't want to lose (notes apps, to-do lists, etc), you can poke around to see if there's any way of exporting that information; otherwise, you'll be out of luck.
SMS and MMS logs, while not transferrable, are in theory rescuable, depending on what kind of smartphone you own, but it requires a lot of legwork on your end. You won't be able to add them to your new iPhone, however; you'll simply be saving them to your computer. There are a variety of different programs available for exporting messages from your smartphones—SMS Backup & Restore for Android appears to be one popular option. As I haven't used it, I can't personally recommend it, but you can always search Google to bring up more options.
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