If you're a Gmail user then I recommend Inbox by Gmail. However, if you use a different email service - or you use several - then I suggest you take a look at Microsoft Outlook, which is one of the best - if not the best - mail apps for iOS out there
But there are plenty of other apps you could choose from.
There are a lot of alternatives that spring to mind (such as WhatsApp, Facebook Messenger), and which you choose really depends on what you family, friends, and work colleagues use. But if I'm going to replace Messages with anything, I'd go for Signal, because if it's good enough for Edward Snowden, it's good enough for me.
Lots of calculator apps to choose from, but my favorite is PCalc Lite, which even comes with an awesome Apple Watch app.
I like the Start app, not only because it's an easy to use alarm, timer, and stopwatch app, but also because it can be used to trigger certain apps (which include Facebook, Fantastical, Instagram, Music, Pandora, Path, Reeder, Rdio, Safari, Sparrow, Spotify and Tweetbot) when the timer reaches zero.
Let me tell you that there are a lot of calendar apps in the Apple App Store, and finding the right one can the tricky and time-consuming.
Three that I've found to be particularly good are, in no particular order, Fantastical 2 (which comes with a great Apple Watch app), Tiny Calendar (which has some of the best layouts I've seen), and miCal (which has some really nice features such as built-in weather and a birthday manager).
Who doesn't need reminding about tasks? While I think that which reminders app people get along with best is a matter of personal preference, I suggest those looking to replace the built-in Reminders app take a look at Todoist (which is great for people who like to collaborate) or Wunderlist (which offers a number of professional features as part of the Pro subscription).
While there are a number of good in-car navigation apps out there. When it comes to general mapping, the best alternative for the stock iOS Maps app is Google Maps.
If the Stocks app isn't doing it for you, then an app like Stock TickerPicker might give you access to the information you need.