Diagnose: there are only three damps
Condensation is water from the air of your own home settling on cold surfaces: misted windows in the morning, black mould spots in top corners, around windows, and behind wardrobes on outside walls. It's the cause in the large majority of UK homes, and the one you can fix yourself.
Penetrating damp is rain getting in: a patch that darkens after wet weather, often at a chimney breast, below a window, or under a gutter joint. Rising damp is rarer than the salesmen suggest: a tide mark on a ground-floor wall, up to a metre high, with crusty salt deposits.
The tell-tale pattern
Location is the diagnosis. Corners, windows, cold walls, worse in winter: condensation. One patch that tracks the weather: penetrating. A ground-floor tide line: rising. A £15 moisture meter helps confirm, but the pattern usually says it all.
Penetrating and rising damp need a builder
Both mean water is defeating the building itself: a gutter, roof, pointing, or damp-proof course problem. Interior paint and dehumidifiers only hide it. Get an independent surveyor rather than a free survey from a damp-proofing company, which has a product to sell.
Beat condensation: make less moisture, move more air
A household of two produces litres of airborne water a day: cooking, showers, kettles, breathing, and above all drying laundry indoors. The fix is a set of habits, all free or cheap.
Lids on pans; bathroom and kitchen doors shut while in use; extractor fans on during and 15 minutes after showers and cooking (if a fan is broken, fixing it is the single best spend); window trickle vents open; furniture pulled 5cm off outside walls so air can move behind it. Dry laundry outdoors, or in one room with the door shut, window cracked, or a dehumidifier running.
Heat steadily, not in bursts
Cold walls are what moist air condenses on. A home kept at a steady low background heat through winter has warmer wall surfaces than one blasted hot twice a day, and collects less condensation for the same money. If you're choosing one room to keep warm, make it the one growing mould.
The morning windows routine
Misted windows are the day's moisture report. Wipe them with a squeegee or towel, wring into the sink, and crack the window for ten minutes. Two minutes a day, and it removes the exact water that would otherwise be soaking into your walls.
Treat the mould that's already there
Once the moisture habits are in place, deal with the visible mould. Small areas (under a square metre): wear rubber gloves and a mask, wipe with a supermarket mould remover spray or diluted household bleach on a disposable cloth, and bin the cloth after. Don't dry-brush it; that flicks spores into the air. Keep children away while you work and ventilate the room well.
Repaint affected spots with an anti-mould paint once fully dry. If mould covers more than a square metre, keeps returning despite the ventilation habits, or anyone in the house has breathing problems, involve a professional (and in rented housing, your landlord, who is legally responsible for treating damp and mould).