Turn on the mixer. After that, turn on the amplifiers. It is possible that amplifiers can be built into the mixer or the speakers. Refer to the documentation for your specific equipment.
Connect a microphone to one input of the mixer. Set its Gain control (usually at the top of the channel strip) to a middle position. Move the Master Faders (usually bottom right on the mixer) to the 0 level. Slowly raise the fader on the microphone channel strip until you hear sound. Ideally the channel strip fader will also be at the 0 position. Adjust the Gain control as needed to achieve this.
Adjust the overall level of the sound using volume controls on the amplifiers. Attach additional microphones and instruments in the same manner. When each channel is roughly balanced by way of the Gain control on its channel strip, use the strip faders to fine-tune the balance of all the sound sources.
Power down the amplifiers before the mixer if this option is available. Turning the mixer on before the amplifiers and off after the amplifiers may prevent a loud pop going through your speakers.