The simplest way to put your Android Phone to Vibration/Silent mode is to hit the Volume Down button (while the phone is on) all the way until you get Vibrate or Silent icon like shown below:
If you don’t want Vibration but Silent mode, then goto Settings->Sound->Vibrate then select “Never”.
After you do that, you will get the Silent mode as shown below instead of Vibration mode.
Also, you can put your phone to Vibration/Silent mode when your screen lock shows on the screen, simply swipe the Sound icon to the left. OR, can also hold the Power button down until you see a screen pop-up where you can also put your phone into Vibrate/Silent mode.
To do the opposite, to get out of Vibrate/Silent mode, it’s the same method.