No Apple ID for me: If you'd rather not set up an Apple ID, you can tap the Skip This Step link in the lower right corner. You can always add or create one from the Settings app later, but note that you won't be able to buy anything from the iTunes Store or set up iCloud until you do.
Once you've set up an Apple ID, you can also set up iCloud on your device. iCloud is an umbrella term for Apple's collection of syncing services, and it allows you to sync your photos, apps, contacts, calendars, reminders, notes, and mail across multiple devices.
Choose to set up iCloud, and you're first asked whether you'd like to enable iCloud backups for your iPhone. If you do so, you can have your device back up all essential settings to your iCloud account; if you ever need to restore, you can do so over Wi-Fi without an additional computer. You can also elect to use iTunes to back up your iPhone to your computer.
Additionally, you're asked whether you'd like to opt in to iCloud's Find My iPhone service. This enables location monitoring for your device, allowing you to find it using your Apple ID and the Find My iPhone app, should it go missing.
One of the iPhone's nice features is Apple's personal voice assistant, Siri. If you want to take advantage of Siri, you can enable it here. Then, once you finish the setup process, you're asked if you'd like to send Apple anonymous diagnostics and usage information (similar to a desktop crash report). After you answer that question, your iPhone will be all set and ready for you to begin using.
Get started with your iPhone 5
Now that your device is up and running, here are a few quick pointers.
Tweak your settings: Most of the underlying system information for your new iPhone--network settings, Mail, sounds, messages, restrictions, wallpaper, and more--is kept in the Settings app. As such, it's a good place to start when you're first getting acquainted with your device.
Set up iTunes sync: If you didn't restore from an older iOS backup, you won't have any music, video, podcasts, photos, or books on your iPhone. You can remedy that by connecting it with your iTunes library: Just plug your device into your computer via its included USB cable.
Explore the App Store: Your iPhone comes with a bunch of cool built-in apps, but you may want to reach outside the sandbox for more interesting fare. The App Store, which you can access on your device or via iTunes on your computer, features more than 750,000 downloadable apps. To explore the store on your iPhone, tap the blue App Store icon on your home screen. Featured and Charts are both great places to start looking for recommended apps, and you can search the entire App Store by tapping on the Search tab. As the iPhone 5 has different dimensions than previous generations of the iPhone, there may also be a collection in the App Store for apps optimized for your device. (Older apps will still run on the iPhone 5, but they'll do so with black bars at the top and bottom of your screen.)
Talk to Siri: If you chose to set up Siri on your iPhone 5 during the activation process, you're just one Home button press away from having your first conversation. Siri can help you book appointments, find restaurants, look up the weather, talk sports, and occasionally whip out snarky quips about robotic AI. To start, press and hold your Home button; the Siri interface will appear, along with a few suggested phrases to get going.
What to do with your old device
Now that you've managed to transfer your old information, set up your new iPhone 5, and start using it, it's time to figure out what you'd like to do with your old device. If you're interested in selling, recycling, or passing it along to a friend or family member, Macworld contributor Joel Mathis whipped up a handy guide that contains the steps you'll need. But what if you're interested in keeping it around?
Turn your iPhone into an iPod touch: No, your old iPhone won't make calls or hop on a cellular network without a data plan, but you can keep using it as a Wi-Fi only iOS device. To do so, you'll just need to keep the phone's old SIM card (or a cheap SIM card from the same service provide you originally used to activate the phone with) inside it. From there, you can restore it to factory settings, and set it up as a brand new device.