1) Developr options: as stated in official site on enable Developer options,
On Android 4.2 and newer, Developer options is hidden by default. To make it available, go to Settings > About phone and tap Build number seven times. Return to the previous screen to find Developer options.
2) Engineer mode: If you comes across an issue that needed to do test, ex: Hardware test, Connectivity test or Telephony Test. Then you will need to get into Engineer mode, all you need to do is: open dial pad -> type in *#*#13411#*#* -> then it will automatically display engineer mode interface.
3) Recovery mode: there are two ways to get in Recovery mode
ii) turn off phone -> press volume down button + power button at the same time -> it will turn on with 3 selections, choose Recovery mode -> note: in this mode, volume up button is Enter/OK , volume down is Navigate/select -> then an Android robot will appear, press power button and you will be in Recovery mode. Refer video uploaded below for clearer steps.
For more details tips on MIUI version 5, check out this thread: [Tips & Tutorials] [Update] 64 Tips You Should Know About MIUI V5