How to back up old stuff onto new iPhone 5?

Just got an iPhone 5 for Christmas, and I need to transfer my old stuff from my old iPhone to my new phone. I already have my contacts cause the guy from the Verizon store did it for me but he said I need to transfer it myself from my computer. I think I have an idea on how to back it up & stuff but I don't wanna mess it up and lose everything haha. So if anyone who's done this help step through step to transfer all my old stuff onto my new phone?

Plug your old iPhone in and click on it when it pops up. Click the back up option. When it is finished, plug in your iPhone 5. Click the restore button and click restore from backup. Click the most recent one and bam.

The Old-Fashioned Way: If you've already got an iPhone-or another Apple device like an iPad or iPod -all of your settings and preferences, contacts, apps, and content are already stored on iTunes on your desktop. Easy! So pop over and download the latest version if you don't already have it. Then, connect your phone to your computer with the Lightning cable that came with it (we'll get back to that cable in a bit), and start syncing. The New- Fangled Way: You can also simply sync up your iPhone over iCloud. During setup, you'll need to connect to Wi-Fi, and select whether you want to set up as a new device, restore from an iCloud backup, or restore from an iTunes backup. Whatever you choose, you simply enter your Apple ID login information (you have to pick just one if you have multiple, though you can change it later). If you're restoring from a backup, your phone will restart, and all of your previous settings will be in place.

See the image in below link

Download a software called Moborobo the new version supports backup and restore your data, I've backup and restored contacts, images, video and music think you can even backup and restore your SMS messages if your device is jailbroken.