The Reminders app in iOS 13 is getting a lot smarter. When you type in a reminder, the app will be able to understand when and where to notify you. Apple is also adding a feature called "Smart Lists" that will integrate with the Messages app when you tag someone.
The Apple Music app will let you scroll through lyrics for music you're listening to in real-time.
The Mail app is getting several changes, including more choices for customizable fonts, multicolored flags for messages, the ability to format messages like you would on desktop, and more.