At this point, for most people, there's really no need to use iTunes if you have an iDevice. For my part, I still use iTunes for listening to audio when I'm at my desk, since I'm used to it at this point and don't want to bother learning something new, but even for me all of the tacked-on iDevice management stuff has always made it feel like a bloated piece of crap, so I eagerly ditched using those to manage my iDevices as soon as it became reasonable to do so years ago.
Backups: Use iCloud Backup in Settings > iCloud > Backup. It automatically backs up in the background when you do your regular charging of the device and have WiFi access, meaning that you never need to remember to do it. The 5GB free option was always enough for my backups, but the upgrade prices are reasonable ($1/mo. = 20GB, $4/mo. = 200GB, $10/mo. = 500GB, $20/mo. = 1TB), and that space can also be used for iCloud's other services, like iCloud Drive (their alternative to Dropbox/Box/OneDrive/Google Drive/whatever that can be used on iOS, Windows, and OS X). If you buy a new iDevice later, all you need to do is give it your iCloud login info and it can automatically restore from the latest backup. Makes getting a new device super simple.
Apps: Buy directly from the App Store app on iOS. No need to ever use iTunes to sync apps.
Music: If you've already made the move to streaming services (e.g. Pandora or Spotify), this is a non-issue. If you're someone like me who prefers to own his own music, then you can use something like Amazon Music, Google Drive, iTunes Match, or a similar service to sync your music from your PC to the cloud, then from the cloud to your iDevice (either with streaming or downloading). I use iTunes Match ($25/year), which, despite its name, only requires iTunes to be used for the uploading process (e.g. the initial upload, then any subsequent uploads of music you purchased from places other than iTunes). After that, you can uninstall iTunes on your PC and continue to use the service just fine. For me, it won out since it's built into iOS (Settings > iTunes & App Store), and once you enable it and tell iOS to Show All of your media, you can play any of those uploaded tracks directly from the cloud (freeing up all of that space on your device), or else download them to keep locally. Amazon Music and Google Drive can do similarly, I believe, though they run in their own apps. I'll let others provide details regarding them.
Photos: Use IFTTT or one of a number of other services that can run in the background and automatically upload your pictures to your Dropbox/Instagram/Flickr/Google Drive/whatever. You'll never need to remember to back them up again since they'll automatically be wherever you want them.
Video: Between all of the sources for streaming video (e.g. YouTube, Netflix) and all of the ways to load your own video onto an iDevice (e.g. VLC), there isn't a need to use iTunes at all. You can still rent/purchase stuff from the iTunes Store vide your iDevice, without ever launching iTunes on your PC, if you're so inclined.
Of course, all of this stuff is specific to iOS devices, so it won't help you at all with the non-touch iPods. I don't have an answer for you when it comes to those.
TL;DR: Ditch iTunes, but keep the iOS device and use third party services or built-in iOS functionality to provide those features. You should be able to do everything just as well or better.