The popular email app Acompli has been available on iOS for a while now, and today has officially arrived on Android as well. It not only handles many of your email accounts, but it also features a Focused Inbox that really looks to keep you on task when it comes to email. You can easily add in your Gmail, Exchange, Yahoo, or Outlook.com addresses.
Acompli bases your Focused Inbox not just on your big interactions, but also whether you actually respond to a particular contact. Focused messages stay apart from "other" mail, but you can reflag (or turn off) whatever you need to in order to keep things straight. You can also choose to receive alerts only for Focused messages, keeping you on track rather than constantly looking at your phone. This all stays within the app so you can still keep things tidy wherever else you check your email.
Add in integration with Dropbox, OneDrive, and iCloud and you have a nice little email app on hand. There's a built-in calendar and people function, plenty of settings, and a clean UI to handle it all.
If it sounds like the email app for you, grab it free from Google Play.