What is "Standby" for?Standby is used to make sure the amplifier is quiet and lower the power dissipation inside the amplifier during times when you will not need it for some period of time. Breaks in song sets are the ideal illustration. If you'll be coming back to the amp within a short period of time (under an hour, say), flip it to standby.
Standby also serves as a softer power up switch. To get the most possible life from a set of tubes, the tube heaters should really be hot before the main power supply is applied to the tubes. When tube rectifiers are used, the rectifiers do this almost automatically. However, when solid state rectifiers are used in an amp, the main power supply comes up almost instantly, and this can shorten tube life by the somewhat-esoteric means of cathode stripping. The standby switch can be used to prevent this.
How should I turn it on and off?
If it has a standby switch, flip this to standby, and then turn the power switch on if it is separate, or if the power switch is simply ON-OFF-Standby, turn it to standby. Leave it this way for 15-30 seconds, then turn it to normal operation. This gets the heaters hot before the main power supply comes up.
To turn it off, simply flip the power switch to "off", don't use standby. This lets the still-conducting tubes bring down the high voltages in the power supply.