It is completely possible to use an iOS device without iTunes. You can buy apps, use iCloud to transfer documents, use other apps (DropBox) to do more, and backup to iCloud.
However, iTunes is still a very useful tool. And it is free. So why not use it? If you had a toolbox with a hammer, wrench, screwdriver and saw, why would you intentionally decide to not use one of them in particular?
Maybe you are under the false impression that you have to buy into some sort of system if you use iTunes? That’s not so. It is just an app. You don’t need to use it for things you don’t want to use it for.
For instance, you could simply use iTunes as a way to back up your iPhone. Or, you could use it to easily transfer music from your computer to your iPhone. But you don’t need to buy music through iTunes or subscribe to podcasts. You can decide how you use it. And since it can be useful to you, then why not at least try it? It is just an app. Just a tool in your toolbox.
Copy Trans Manager apparently allows you to manage your iPhone without using iTunes:
Alternatively Mobyko is a service that many of my friends use (available worldwide, just checked)
Uptact is a Contacts Sync App that will keep your Contacts Updated