IMO, the Salesforce Mobile App is the one thing that separates Salesforce from all other basic CRM systems - it is probably the ONLY reason you would use Contact Manager, Essentials, or Group Edition of Salesforce. 

The new Lightning Mobile App looks great, but it takes a lot of setting up! So beware. 

Firstly I would NOT start with the in-built tool to make everything in your org Mobile Enabled, unless it is a very simple org. It's too much. I may even decide to create a simplified Mobile app that includes ONLY the tabs that are helpful on Mobile. I used it, now I have to go back into every page and turn off Mobile so that I can turn off certain Apps to be available on Mobile. 

The old Mobile App included EVERY single Tab that you have access to - there does not seem to be a way to turn that off. That is not good. As a System Administrator I would love to have a Mobile App that is just some key details that I would like to keep an eye on on Mobile, not the keys to the kingdom. But even if you create a simple App that only has a few tabs, you can still go to All Items. So annoying. I want nothing on Mobile unless I say so. 

Edit each and every Lightning App

Edit each and every Object

Page Layouts

Lightning Pages

Home Pages

Favorites

Roll Out

There is a whole Trailhead on Planning your Roll Out Strategy and talking about Micro Moments and how to craft your mobile App to best suit the needs of your users. Sure, but if you can't hide MOST of the things that they can see in the main app anyway, how can you craft something that's mobile specific. You're just going to frustrate your users. 

Navigation Settings

What you can See

The Data Access and Views page is probably the most helpful to understand what is going on. Actually go through every document on this page, except there is NO documentation similar for the new Salesforce Mobile App, which is completely ludicrous!. 

Other Features

Salesforce Today

Oh what a complete waste of time feature. This page says it all. After reading that, I don't know why you would even bother with touching it. They basically say, it's broken, use it at your own discretion. 

Security

There are some features you can set on the Connected App, but there is much more you can do to lock down Mobile Security - IF you pay for it. This is not good. Security should not be a paid feature. 

I don't know how long this has been around, but you can now require a pin on the mobile apps - so that means you can use fingerprint authentication if your phone supports it. However I can NOT get it working on Android for fingerprint AND it does not support Face login for Android at all. But make sure ALL users' phones support it, or they will be annoyed with a pin and won't use the app. 

The app now integrates with 1Password and LastPass - again, really good. 

Offline

Does it work? Who knows... here's the docs. Read them. And note What's Available Offline.  

You can have 7 objects to access for Offline. Would you even need 7? I picked 4 key objects. 

If a user chooses not to customize their cache, Salesforce populates the user’s cache with up to 30 recently accessed records for their five most recently accessed objects. In addition to these records, the user’s tasks listed under My Tasks and their five most recently accessed dashboards are cached. However, if the user chooses at least one item from the Offline Preferences page, this selection replaces the existing cache with their new preferences.

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30! Is that all?!

Then remember to start the Cache. 

TIP We recommend that your users populate their cache each time they log in to Salesforce so they have a meaningful set of available data when offline.

To me, something as bizarre as this means that they KNOW it's not working. You only need offline WHEN you are offline, you rarely plan to be offline. 

Notifications

The new Notifications are really cool. You can name them, and the names of the notifications appear in the notification options and users can decide to turn them on or off. If you need Full Content Notifications, set that in the Connected Apps, but beware of your data privacy when you do that. ALSO remember to got to Salesforce Notifications menu and turn both options on, then go to Notification Delivery Settings and at the bottom of the page, choose your Custom Notification Type, click Edit in the drop down box, then choose your IOS and Android Apps. This also then ads a section in the Push Notification Settings in the mobile app where users can turn these custom notifications OFF!