(phone) Settings > Notifications
(i.e., pull down top shade, tap gear icon)
Choose the app you want to customize.
(Let's use Calendar as an example)
Choose Notification Categories
Be sure Calendar Notifications is toggled ON (right side of line)
Okay, now ... on the LEFT side of Calendar Notifications line, tap the Left side of that line, where "Cal" text is
You are now at the Calendar Notifications custom detail screen
Tap on Sound to change the sound you want.
It's the same procedure for other apps.
If you want each person in your Contact list to have a different ring tone, it's slightly different, but follow along ...
Open the Phone app.
Tap Contacts tab (bottom), tap on the person you want to change the ringtone.
Tap Info icon (round circle with "i" in center - to see their contact details), then scroll down to bottom
Tap Edit, and scroll down again to bottom
Tap on the ringtone line
Change the ringtone to what you want