Let's face it, the apps that come preinstalled on your iOS device aren't always the best. There are many more capable, full-featured apps on the App Store that can replace them.
Fantastical 2 Calendar Replacement $4.99
Fantastical 2 is the best calendar app for iOS. It syncs with all of the calendars on your device (iCloud, Google, Holidays, etc.), has an extremely powerful today widget (displays an actual calendar view unlike the preinstalled calendar app's today widget), a beautiful user interface, and allows the user to create new events using natural language instead of filling out a form.
Google Chrome Safari Replacement
Safari for iOS is a great web browser, but sometimes you just want to try something different. If you are ever in the mood to change up your web browser, you should try Google Chrome. Chrome has a very friendly user interface that is bathed in material design, Google's design language. It also contains an interactive tab view like Safari so you can easily see what is going on in all of your tabs.
Notability Notes Replacement $5.99
Notability is an extraordinary note-taking app for iOS; it even beats the new Notes app in iOS 9. Notability features exceptional note organization by allowing users to group "Subjects" of their notes within separate "Dividers." This is very useful for keeping your school, work, and home notes apart and categorized.
Google Maps Maps Replacement
Google Maps is Google's well trusted navigation and mapping service. It boasts driving, transit, walking, and biking directions in its app, which is more forms of transportation than Apple's preinstalled Maps app offers (it lacks biking directions). Google Maps also has generally more accurate data than Apple Maps, which infamously had many major mapping errors when it was first released a few years ago.
Microsoft Outlook Mail Replacement
Microsoft Outlook is by far the best third party alternative to Apple's Mail app. Outlook can send files from many cloud storage providers (such as OneDrive, Dropbox, and Google Drive). These files can be sent without even being downloaded to the iOS device as well, which can dramatically save device storage depending on the file size. Outlook also has a built-in calendar so you don't have to switch apps to manage your schedule while you're checking your email. If you want to try Outlook, don't worry, it's not just for people with Microsoft-hosted emails. It is compatible with all major email providers.