The Xiaomi Redmi Note comes with the feature-rich MIUI Browser. It does a great job browsing, syncing with your Google account, supports downloads, there is a night mode, too. It is as fast as Chrome, but does not support Find on Page or copy text (only links). On the positive side, it has a built-in Reading mode, which isn't available in Chrome. It enlarges the text font and strips away unneeded elements, leaving only the article you are reading in focus.
Unfortunately, the MIUI Browser didn't handle pressure particularly well - it couldn't handle our battery-test browsing script for more than three minutes without crashing.
That's why we suggest you install Chrome, at least as a backup, just in case you need to avoid crashes or you need to find something on a page. Plus it can sync with your Google account easily.
The Xiaomi Redmi Note offers a great file managing app called Explorer, but there is no document viewer/editor pre-installed. There are plenty of free options in the Play Store (even ones made by Google), so this shouldn't pose a problem.
Xiaomi Redmi Note also features the WPS Office suite, which supports creating, viewing and editing office files (docs, xls, ppt).
The MIUI v.5 also offers a useful Security app. It can scan your phone for malware, manage your blocklist, manage or restrict your data usage, configure battery behavior, clean some RAM, it can also manage the permissions of your installed apps.
The custom Calendar looks good, syncs with your accounts including Google, and offers Day and Month views.
There are also the standard sound recorder, notes, flashlight, calculator, clock, and weather apps, among others, that are a given in any self-respecting Android package nowadays.
By the way, you can activate the torch from the lockscreen by a tap-and-hold on the Home key.
When it comes to getting around, the Xiaomi Redmi Note relies on Google Maps and Navigation. Naturally, if Google Maps isn't installed on your unit, you can get it for free from the Android Market. The app offers much of the same functionality as its web-based counterpart, although you will need a data connection to take full advantage of the navigation features. Street view mode with digital compass enabled is an especially neat thing that you should definitely check out if you haven't already.
Whatever you may be missing, you'll find it in Google's Play Store.