With convenient inbox management tools for all your accounts and a built-in calendar, Microsoft's Outlook for iOS is a great option for email and appointments.
Open the slide-out menu in the upper left to quickly switch accounts and check specific folders.
Swipe left on an email to archive, or you can swipe right (pictured) to schedule that email for later in the day when you have time for it.
You can easily create a meeting using the calendar and even set up reminders for yourself.
In this view you can see your day hour by hour and your meetings and appointments.
Outlook organizes attachments by email account, and you can add services like Dropbox and OneDrive depending on where you keep your files.