Mould needs three things: spores, an organic food source, and moisture. Spores are in every building and cannot be removed. Food sources — dust, soap residue, drywall paper, grout — are largely unavoidable.
Moisture is the only one you control, and everything effective follows from that.
The fan, and the twenty minutes after
A fan that runs during the shower removes some moisture. The room continues evaporating for a long time afterwards, from the walls, the floor, the towels and the shower itself.
Run the fan for twenty minutes after the shower ends. That single change does more than any cleaning routine.
A timer switch or a humidity-sensing switch does it without anyone remembering, and the humidity sensor is the better of the two because it responds to the condition.
Wiring the fan to the light switch guarantees it stops when the light does, which is exactly the wrong behaviour. Separate them.
Where it grows first
Grout and silicone, which hold moisture and soap residue.
The ceiling above the shower, which is the coldest surface in the room and where warm moist air condenses first.
Behind and under the toilet cistern, where cold porcelain condenses in a humid room.
Around the window, on the frame and the reveal.
Inside the fan housing and duct, where nobody looks.
Under the vanity, particularly where there has been a slow supply leak.
Each of those is a place where either water sits or a cold surface condenses, and both are moisture problems rather than cleaning problems.
Silicone against grout
Grout is porous and it is what mould grows in. Epoxy grout is not porous and resists it, at higher cost and more difficult installation.
Silicone goes in the movement joints — inside corners, and where tile meets the tub or pan — because grout cracks there. Mouldy silicone cannot be cleaned effectively; it is cut out and replaced, and that is a routine maintenance item rather than a failure.
Use a sanitary grade silicone containing a fungicide in wet areas.
Habits that help
Squeegee the shower after use. It removes most of the water that would otherwise evaporate into the room.
Leave the shower door or curtain open afterwards so it dries.
Hang towels spread out rather than folded, and outside the bathroom where possible.
Keep the door open after showering, unless the extract is running well.
None of that is a substitute for ventilation, and together with it they change the drying time substantially.
Things that do not work
Bleach on porous surfaces. It removes the colour and leaves the growth, so it looks clean and returns. On grout and silicone the growth is inside the material.
Painting over it, which lasts one season.
Cleaning without changing the moisture, which is treating the symptom on a schedule.
Mould-resistant paint as a solution rather than a small help. It slows growth on the surface and it does not address why the surface is wet.
When it is more than surface mould
Growth that returns within days of cleaning, growth in more than one part of the room, a musty smell with no visible source, or staining that spreads on a ceiling all suggest water rather than humidity — a leak in the wall, a failed shower pan, or a duct dumping moist air into the ceiling.
At that point the question is where the water is coming from, and cleaning is not the answer to it.
The duct in the attic
A fan ducted through an unconditioned attic without insulation condenses inside the duct in winter. That water either drips back through the fan or soaks into the attic.
Insulated duct, sloped toward the outside, terminating through a proper cap with a working damper. Those four details prevent a problem that presents as mysterious drips from the fan grille.
Windows are not ventilation
An openable window is useful and it is not a substitute for extract. On a cold day nobody opens it, and on a humid day the outside air brings moisture in rather than taking it out.
Several jurisdictions accept a window in place of a fan in the code. That is a minimum standard rather than good practice, and any bathroom with a shower benefits from mechanical extract regardless of what the code allows.