Backup iPhone contacts etc to move to new iPhone
iCloud is your most hassle-free bet. On your 5s, go to Settings > iCloud. You'll find the following settings.

• iCloud Drive: Any apps turned on here may store content in iCloud without the use of iCloud Backup. This means when you get your new phone, you can set it up as new and still have access to the same content if you redownload the apps.

• Photos: This one's a bit trickier. You have switches for both iCloud Photo Library and iCloud Photo Stream. These are two separate things and the only reason to have both on at the same time is if you have devices on both iOS 8 or later and devices prior to iOS 8.
iCloud Photo Stream is available starting with iOS 6 (I believe) and only stores your most recent 1000 photos. It does not keep track of which photos are in what albums, does not keep track of edits, and when you get your new iPhone it will only download the most recent few hundred. iCloud Photo Stream does not take up your allotted iCloud storage space, which is 5GB if you're not paying monthly for more.
If you're willing to take up more of your allotted space, assuming you have enough, iCloud Photo Library is a better option available on iOS 8 and later. It has no limit on how many photos it can hold, also stores videos, keeps track of edits and albums, and allows your new phone to get all of the same content with no limits.

• Mail: Stores emails from any iCloud email accounts. If the accounts are not email, you will likely still be able to get your stuff on the new phone when you sign in again.

• Contacts, Calendars, Reminders, Notes: Stores the associated app's content without the need for a backup.

• Safari: Stores bookmarks, reading list, history, iCloud Tabs, etc.

• News: Stores information on what you've subscribed to and liked/disliked.

• Wallet: Stores passes, cards, etc.

• Keychain: Securely stores your saved passwords from websites and apps to the cloud.

• Backup: Everything above will work seamlessly without needing a backup, and when you get your new phone you can just "Set Up as New" and it's all automatic. If there is some stuff you want to keep not covered above though, such as an app not supporting iCloud Drive, you can turn on backup and in Settings > iCloud > Storage > Manage Storage > your backup you can choose which apps specifically you want backed up. If that app is already turned on in a step above, it probably doesn't need turned on in Backup.
If you use Pages, Numbers, or Keynote make sure Use iCloud is turned on in Settings > Pages, Settings > Numbers, and Settings > Keynote.
If you're using Backup, when you get your new phone you'll need to do "Restore from iCloud Backup" instead of setting up as new.

• If you don't have enough iCloud space, you can always experiment with other clouds like Dropbox, but iCloud is the most seamless. There's also iTunes Backup like you mentioned which is easy but not quite as seamless.