As you know our RN3 come with Locked Bootloader so first of all you have to unlock bootloader via official method trust me its a headache you have to wait around 7 days to get unlocked if you are lucky enough. In this tutorial i only covered how to flash twrp recovery.
for unlocking help visit official thread http://en.miui.com/thread-202290-1-1.html
ADB Driver if Device Not Detect
How to Flash Twrp Cofface Recovery in Redmi Note 3?
1. Install xiaomi usb drivers for Redmi Note 3
2. Extract rn3recovery.zip
3. Turn off your device and hold volume down and press power button.
[Now you are in fastboot mode]
5. Plug the device into your computer.
6. Now Double Click on install cofface recovery.bat name file inside rn3recovery folder.
7. You Recovery will be flashed but wait do not reboot your device.
8. Hold Volume Up button and press power button to successfully boot into twrp recovery.
How to Root?
1. Download SuperSu from link given below.
2. Boot into TWRP recovery
3. Select install and choose supersu zip
4. swipe to execute and you will have rooted device.