I have sound coming into the mixer but nothing coming out of the speakers or monitors.

On the back panel there is a section labelled Patch Bay
There are 2 small switches which assign each Power Amp to either Monitor or Main Output.
It would seem you have them both set the same at present. Set one to Monitor and one to Main, and use the corresponding Power Amp outputs direct to speakers.

Note you can send signal to external amps or powered speakers from the preamp outputs. This is really handy.
Download and print the manual from https://www.manualslib.com/manual/90516/Kustom-Kpm-8420t.html
and keep it with the mixer.

( Of course don't send speaker outputs to other amps or powered speakers )

An external equalizer will not help this at all. The problem is likely within the amp (feeding the main speakers), the connections from the board to the amp, or the speakers themselves. In other words, the answer would be "NO." Best of luck!

The outputs of these mixer needs to be connected to an active speaker.

An active speaker have an amplifier and a preamplifier.

Depending on what other settings are chosen- this at times can be normal. You may be simply switching from the output of the mixer board to the sound input source, and if gain is up and mixer level is up, you will hear it. If it is outputing to the host and it should not be; check stored settings and Scene memory.

If the mixer is receiving signal, then the problem is pilot error at the mixer, not the guitar. You did not mention what type of mixer you have so I have no way of helping. Common problems would be failure to send the guitar channel to the mixing buss, leaving the channel muted... but we really need the mixer and model number to give more help that that.

The Efx 12 has two outputs monitor out as well as mix out ( XLR sockets ). If the level bars are moving its an indication that the mixer is working. Have you tried headphones out as well as monitor out.
What is your amplifier and speaker setup- you have not mentioned it at all.
Even if any PFL switch is on you won't get sound out of the speakers.
PFL stands for Post Fade Listen and when any of the 12 channel PFL switch is on only the headphone out put will work.

It is likely you do not have the levels set correctly and are over driving the mixer. Reduce the trim settings using the PFL buttons and monitor the level indicators to insure you are not clipping.

With a condenser mic you CANNOT have any speaker in the room that the mic can hear of course. These mics are so sensitive. Your mixer and monitor speakers and recording will be in a separate room from the soundproof studio typically.

To monitor the system anywhere in the same room as the mic you MUST use headphones that do NOT leak sound or you will get feedback and distortion.

You will need at least a 24 channel mixer to give you spare channels. You also need an active crossover and a snake. The advice you need is too much to include here so I am including some links:http://mixonline.com/live/FixIts/, http://en.wikipedia.org/wiki/Live_sound_mixing, http://mixinglivesound.com/blog/tag/outdoor-concerts/,http://mixinglivesound.com/blog/, and http://www.scribd.com/doc/11435651/Beginners-Guide-to-Mixing-Sound. You may need more monitors than you are planning on and you might consider in ear monitoring.

You don't specify which model Phonic mixer in question.
Is it a powered mixer ? And are you plugging your speakers into the speaker outputs ? If it's not a powered mixer and you're plugging speakers into the Line Outputs, you're not using a power amplifier between the line outputs and your speakers - this won't work.
If it is a powered mixer and you are plugging speakers into the speaker output connectors and still getting little or no sound, check your speaker cables and the speakers themselves by plugging them into known good amps.

try connecting speakers to another sound source or if you have access to a ohm meter, check impedance of speakers, should measure 4~8 ohms.