Sounds extreme, but it's not harder than for example skiing. Moving human body generates a lot of waste heat (my estimate is around 0.5 kW), all we need is to keep it inside.
Firstly, it's good to expose as little skin as possible: long sleeves, gloves and winter hat are indispensable, glasses or goggles are nice to have, and a mask or scarf over your face helps to preheat the air for breathing. Waterproof clothes are not needed because snow doesn't wet you, but windproof top layer is very useful.
Secondly, very much depends on how far you are going. For just a few kilometres all you need is a light windproof jacket over more or less ordinary clothes, because getting cold takes longer and you get enough heat from pedaling. If you don't overdress, sweating is just an option. For longer trips, on the other hand, clothing must be selected carefully: it must insulate even when wet (sweating is certain), it must adequately cover all sensitive spots which take longer to cool down (knees, forehead, feet etc.) and you need to be able to survive a few minutes' stop. Cotton is not up to the task, functional fabrics are needed, either synthetic or wool. And be sure to pack a warm coat for case of unexpected longer stop (flat tyres and the like).
Lastly, adaptation is an important factor. The best way to adapt to cold is to not quit biking at the end of summer and just continue as usual. At the end of September it's like: 10 °C, yuck, so cold. By November the definition of cold shifts to five degrees, 10 °C feels warm. In December the cold begins somewhere around the freezing point and it keeps shifting like this until you find yourself riding in T-shirt in March, complaining about scorching hot four degrees :-). Of course, everyone's comfortable temperature is different. Important is to not overdo anything and always have some extra clothes in reserve.
And some real-world example? Let's take my usual 80 km trip, ridden until frost or snow comes (slippery country roads in the dark are not safe). The following equipment works fine for me: two pairs of socks (one of them wool), two trousers (wool bottom layer and synthetic elastic top), long sleeve moira shirt and warm jersey, quite ordinary fleece gloves, moira hat, knee sleeves and a folded hat in my crotch (don't laugh, I have seen how bad thing a testicle inflammation is: half year of pain and a permanent damage as a bonus). This applies to recumbents where feet and butt get the most wind and hands and head stay relatively warm. On upright bikes, the most exposed parts are hands, knees, forehead and ears, so I'd take warmer gloves and maybe a second hat or headband. To unfreeze my numb feet, I keep pacing back and forth during each "refuelling" stop, and sometimes get off and push up a hill.
Unless you're a fan of ice-cold drinks, you need to somehow keep your water at acceptable temperature. Vacuum flasks are too big on the outside and too small on the inside for my taste, so after several experiments, I ended up with the pictured "thermo" bottle: one-litre outdoor plastic bottle packed in two thick socks and a plastic bag, hidden in a pannier. Before departure, I fill it with water hot just enough to not scald my probing finger or melt the plastic. After 20 km it is hot but readily drinkable, body-warm after 40, and about room temperature when I finish the last third at 60 km.
Footsoles are the most windward parts on a 'bent and because my light SPD shoes are far from winter-rated, I tried to hide them behind streamlined foam blocks:
The result: dead end, not worth the trouble. I think the part of my foot directly over the metal cleat was less cold than before, but it might be just a placebo effect. The cost was significantly more difficult clipping in and out - not something I want in a city. In the end, I got much better insulation with no side effects by putting on a second pair of socks.
Thin layer of new snow is harmless, it just slows you down a bit. Thick layer would stop you, but we are not in the middle of wilderness, city streets are usually cleared. Cyclepaths are not that lucky here in the Czech republic, so after a snowstorm, you are usually limited to choose between a salt-covered main road narrowed down by snow banks to the width of a typical car, and ankle-deep white mayhem everywhere else. Uncomfortable, but survivable. Don't mind the honking - on the second day, the drivers realize you really mean it, and stop (my experience).
Intermittent snowing and temperature oscillating around freezing point are just the right prerequisites for slippery roads: the snow melts, then freezes again and you get an uneven layer of ice. That's bad. Ride slowly, watch the road and pick a path least likely to skid your front wheel. Hardcore level: ice masked by fresh snow. Luckily, ice like this only stays on back streets with minimum traffic, main roads are usually salted down to the asphalt.
Salty slush is not outright dangerous, but it sticks to wheels, clogs mudguards, slows you down and corrodes unprotected metals. It's good to let your bike drip and dry indoors after each ride, and lube the chain frequently.
When the temperature drops deep below zero Celsius and stays there, cycling becomes surprisingly safe. Ice and slush have probably been cleared away earlier, new snow is an inert powder not more slippery than dry dirt, skids can be recovered from and falls avoided. So just don't freeze to death and everything will be fine.
Overnight warm-up and rain may stain the frozen ground with "black ice", the most slippery stuff ever (see picture). Asphalt looks like wet, but you will be shown the difference very quickly. With some skill and luck, it can be ridden, but only straight or downhill. It's safest to get off, push and use the bike as a balance aid. Black ice melts away from salted roads quickly, but can stay longer on untreated sidewalks.
Last few random remarks. Frosted rails and speed bumps are even more slippery than usually, so be careful and cross them perpendicularly. The little sharp rocks intended to make roads less slippery are also pretty effective at making bicycle tyres more flat. It's good to practice and test your adhesion limits somewhere on a frozen pond, far from cars and other dangerous objects. Wider tyres at lower pressures grip better than hard slicks. Studded tyres exist; some fellows praise them, I haven't tried them yet.