Your wall plate should have a power source cable, your ceiling light & fan cable, and lastly your vanity lights. All this cables should be Black, White with a Ground wire. If you have a cable with a Red wire put that one to the side for now (likely your fan unit) we’ll deal with it at the end. If you have more cables these likely serve your other lights and outlets in the basement. This is were you need to stop as you likely have to much load on the circuit. Bathrooms always should have their own circuit and also need GFI breaker and outlets. Your basement outlets likely need a AFCI breaker. This is at a point you need a licensed electrician to install a new line to your bathroom and rewire the basement service so the line is not present in your switch plate gang
You’ll need to use your volt meter or voltage detector probe to identify your power source cable. I would recommend you get some colored electricians tape so you can mark it with color (Yellow). Now carefully tie all of the neutral wires together (While) cap and tape the wires so nothing is exposed. Now with the power feed Black wire isolated so it doesn’t short out carefully touch one black wire to it so you can identify what its powering. WARNING! Use extreme care with the live wires as you don’t want to get a shock! Once you know what each cable does mark it with a different color tape and make note each colors function.
Now lets talk about that cable that has the red wire, now that you have identified all of the other lines this should be your ceiling fan. The fan and the light are likely setup so you can use each independently from each other. So you will need to test the Black & Red wires like we did the back wires one wire will turn on the fan and the other will turn on the light. I wired up my fan with a timer so set it and walk away without having to worry to turn it off.
Depending on how may switches you have and if you are using a high density switch unit you might consider placing the ceiling light and fan control outside of the bathroom saving the space in the other box gang for an outlet or night light.
Now your switches need to be laid out in the order you plan to put them into the box. with the breaker turned off wire up the black lines to the switches keeping them on the same side (top or bottom). You’ll need a black wire tail to mount on the other side of each switch. Carefully strip off the excess wire on the tails so you can tie them together and take the source Black wire bind it to the other black wires. Mounting the switches is all there’s left.
Well this is not what I was expecting!
So you'll need to test the lines from each cable pair. Right now you have the cris-crossed!
Place the Whites up and the Blacks down. The left cable wire group away from the high cable wire group. Cap the Black wires as thats your hot wire so you don't get zapped.
Use a voltmeter to test the pairs one will be live the other not. That pair is the power feed for the switch. Now the next part is a bit tricky!
The other cable feeding either your light or fan directly and then within the fan or light you have another cable interconnected that feeds the other.
You only have one switch possible here, the two Black wires for to the switch and the two Wire wires are tied together as outlined here How to: Wire a Light Switch. You would need another wire from the fan/light to offer the needed connection to power it independently (the Red wire) and if you have a vanity light that would need its own box and would be setup as the diagram independently.
I think you'll need to get an electrician in to re-wire the fan/light with the needed cable and look at how the vanity is wired.
@stormcrow83 we'd need to see the "double switch" and the wiring on your vanity and ceiling light. There should be a junction box somewhere that might show you what is going on. There will be no bypass but it sounds like somebody tapped into the circuit to your basement etc. How old is your house and where in the world are you located?