Here are my top recommendations for email apps for your iPhone.

Outlook

Definitely worth a download, it can handle emails from: googlemail, outlook, exchange, gmail, Yahoo, iCloud and other IMAP emails. I use it for a couple of clients emails and far better than the native mail, it does everything you would want from a mail app and supports custom swipe gestures, allows calendar management (directly within the app rather than using another app) and file management and attachments from a series of cloud services (namely OneDrive, Dropbox, box and Google Drive), uploading files is seamless.
One feature which makes me love the app over others is the focused inbox which you can either educate or will filter emails for you, especially those emails you set up for daily notifications and rarely do anything with. Notifications are completely customisable and very easily changed. I definitely recommend this as your primary email app/mobile service.

Link: Outlook for iPhone email app 

Spark

I actually stumbled across Spark looking through the app store for something completely different.
Spark is great; allows customisation through gestures and badges. The small but underestimated instant replies with like, smiley face and thanks within an email (think quick responses or replying from small devices like the Apple Watch) is really well thought of and is a favourite of mine.
Spark splits out notifications and has a smart inbox which tells you what email is important and the others emails that are generic and not important. It is smart and learns from your activity, personally I prefer this from the inbox app from Google and whats better it supports Google, Exchange, Yahoo, iCloud and Outlook emails. Similar to outlook app its connected to cloud storage providers – Dropbox, Google Drive, Box, OneDrive. It offers read it later services like Readability, Instapaper and Pocket. Spark also supports note taking notes like Evernote and OneNote. If you are like me and want to split out email definitely download this and use alongside Outlook.

Good video explaining its features –

Spark Tip #5: Workflow of pins, archive and snooze from Readdle Inc. on Vimeo.

Link: Spark email app

Inbox by Google

When inbox first came out I spent a few weeks using it and testing every little feature, to be honest I have a love hate relationship with it, I like the layout, the feel and what Google have tried to do it make emails actionable but it just does not understand email the way normal and non Googlers really use email. I like like how you can manage emails, by swiping on each email and marking as done or snoozing emails. It’s great at previewing attachments at a glance and bundling emails; so if you buy a lot through Amazon or have trips its great at bundling together. For when you are looking for an email pinning emails works well as does the search (as you would expect from Google). Something to keep in mind is if you use Google religiously or rely on Google (and love its personalisation) this would be good for you and allowing Google to control your mailbox but personally it just is not as good as Outlook or Spark.

Link: Inbox by Google email app

Notable mention should go to mailbox, it was the first app to make email “actionable” but I never really got to personalise email enough to make mailbox work for me, the mailbox for mac is also a decent app and worth a go if you like the mailbox app for your iPhone.

Here is a good review by The Verge on Outlook app –