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You have two options in here to backup your iPhone, so that you can restore that backup to your new iPhone. First is "To iCloud" and second is "To iTunes on your computer"
To iCloud :
On your iPhone, go to Settings. Tap iCloud. Tap Backup and tap Backup Now button and this will backup the iPhone to iCloud (online)
To iTunes on computer :
We can backup the iPhone 5S to your computer using iTunes
First download and install latest iTunes on your computer from here --> http://www.apple.com/itunes/download/
After downloading iTunes, make sure you install it by opening the downloaded file and following onscreen instructions. After installing iTunes, open iTunes. Then connect the iPhone to your computer. iTunes will recognize it. Then select your iPhone icon at the upper left, next to Musics, Movies etc icons (small iPhone button)
(When you connect the iPhone 5S to your computer and if it ask you to restore from backup or setup it as new iPhone, select setup it as new iPhone)
This will bring the Summary screen of iPhone. Now, click Backup Now button and make sure "To This Computer" is selected there. This will backup the entire iPhone to your computer.
Now, when you get the iPhone, you need to restore this backup to your new iPhone. So everything will be transferred to it. For that, you can follow the steps below;
From iCloud :
While setting up the new iPhone, it will ask whether you want to restore your backup from iCloud or iTunes on computer. So if you choose iCloud to backup, then select Restore from iCloud. But if you backed up to iTunes on your computer, then do the following.
From iTunes on your computer :
Choose Restore from iTunes Backup when you setup the new iPhone. Then connect it to your computer and open iTunes. Then this time click Restore Backup button and from the list of backups, select the backup of your old iPhone from the drop down menu. Click Restore. That's it
If you have any difficulty on following the steps above, please let me know and I will walk you through it.