The moto email app is called "Email". To make it work properly with Active Sync, you have to manually set up the account. You have to set it up as "Exchange" or "Corporate" depending on the app (Exchange for GMAIL or Corporate for Email).
You also have to "Manually set up the account." You can use the default settings when you manually set up the account except for the server. For the server, one of the following should work:
blu403-m.hotmail.com OR s.outlook.com
I prefer the Moto "Email app" over Gmail. The Moto email app will also sync contacts. I think it also syncs calendar pretty well, but I don't use the calendar so I'm not sure.
Even though the new updated Gmail account supports Outlook.com, if you don't do the manual setup it will connect it as IMAP and I do not believe IMAP syncs calendars or contacts with Outlook.com.
I also caution you that the contact sync can do funny things sometimes if you have the same Outlook.com account set up in multiple accounts. Best to only connect your Microsoft account in a single account -- Gmail, Email, Nine or another app.
For everyday users, I think the Moto Email app works just fine.
If you need something more powerful, then I suggest Nine but you have to pay for it.
Stay away from the official Outlook app! It's a POS!!!
Mobile - Outlook.com Help