changing notification sound for different notifications
Open Settings.

To change the system's notifications, navigate to General > Sounds:



Currently there is no way to set apps' notifications unless there is a setting within the app. I recommend you thoroughly look through the apps' settings if you want to do this.

Edit: In regards to our conversation in chat about this topic, here's some more info.

System apps, again, you can change using the above method. Other apps depend on a per-app basis: It is entirely the developer's choice whether to support functionality for customizing notification tones. Facebook, for instance, does not.