The Solar System
The first thing that would happen to Earth, immediately after the sun disappeared... would be nothing. It takes a bit under ten minutes for light from the sun to hit Earth, so it would look like the sun was still there, for a while. Earth would even still be in the sun's gravitational field would also last some minutes, as the sun's gravity takes time to reach us.
Once we were hit by the most beautiful night sky in human history, the Earth would would be flung out into space, away from our solar system. From this clear view of the sky, we could easily see Jupiter and the other planets--sitting in the sky like nothing had happened. The sun's light is still reflecting off of them, and being further away they have not yet been "hit" by the lack of gravity.
The exception to this is Mercury and Venus, which being closer to the sun, have already gone through our dilemma. We probably won't be able to see them anymore, without the sun's light reflecting off of them (maybe we can with certain telescopes).
On the bright side, it'd actually take us forever to suffocate. Sure, all the plants die in a few days, and we have no way to recycle carbon to oxygen, but the atmosphere will stay mostly breathable for many centuries. Though, if we survived any length of time, the breathable air around cities could be pretty bad, without any parks or plants to regulate the local oxygen.
Many trees would survive for a year, before the growing cold killed them.
Within the first week, temperature would hit 0 Celsius, freezing, and you could walk across the frozen sea. That would be the good times. Many people would die in the following weeks from the sudden temperature shift, the poor in hot countries in particular would soon be freezing and sickening.
The Earth itself would become sick with the sudden temperature change, and terrible blizzards would rage across the world. The storms would eventually cease, maybe after the first month, and the weather would become much more stable across the globe.
In fact, when the last storm finished... there'd be no wind. Without the sun, wind would become a local phenomenon.
Snowy weather would become more and more rare as time passes. Without the sun to evaporate water, the only source of evaporation would be geothermal springs. And without the sun to heat the clouds and change temperature, they'd mostly stay up there in the cold sky; supercooling well past the freezing point.
A year later
After a year, it'd get real bad... Temperatures would hit -100 degrees Fahrenheit. The coldest temperature I've been able to find for human habitation was a research post, at -97 degrees Fahrenheit. A couple of cities have also survived winters of -90 Fahrenheit, though I don't know the details for how well they could keep that up.
While those equipped for it could survive, a huge portion of the human population would die. Every plant on Earth would die. Most of the animals across the world would die.
The animal world
For the first few months, the animals in the Night-half of the North Pole would notice no difference. They might get hit by the weeks of storms, but otherwise life would remain unchanged. By the time a year has passed, though, they would've reached a temperature on par with their coldest ever winters, and most of the animals would soon die. It is unlikely, but possible, that some animals will manage to survive for a while longer... only to die of starvation, as what little plantlife there is cannot be renewed.
Surprisingly, insects like the Red Flat Bark Beetle would be among those that survive the cold. They would eventually die out all the same, as their food source (wood-eating beetles) succumbed to the cold. Even if their food source survived, the dying trees would eventually be eaten away. The Upis beetle would outlive the Flat Bark, by freezing--going into cryogenic hibernation.
Cockroaches and rats would also survive in human compounds, though they might disappear entirely from some cities.
Animals who live near hotsprings may survive and adapt to this new world for a while... only to die when the death of plants and eternal winter causes them to starve.
To survive the cold, you either need insulation and power, or hotsprings.
The easiest way to stay warm is to dig deep underground (which will get harder as the cold freezes the ground). Underground living will become quite comfortable, losing its chief disadvantages. Underground, you have to worry about rain, and sewage management. But in this scenario, all your sewage will be recycled, and there won't be a problem of rain anymore. Underground living will help you keep warm and safe, with minimal power consumption. You will, however, need to go to more effort to ventilate your air.
To survive starvation, you need either large food stores (which will run out), or a lot of power and a meticulous setup.
With power and insulation, it isn't too hard to keep warm, and even ventilate your compounds. The real drain on resources will be cultivating food. Mushrooms can be cultivated easily enough, and will begin to make up a large part of your diet as an economic food. But generating enough light to grow crops will be challenging. Without crops, most farm animals are useless.
So largely you'll be eating mushrooms, and vitamin tablets, and eventually lab-grown yeasts and the like to make up for the dietary deficiencies. The most obvious and important deficiency is vitamin D, as you no longer have the sun.
What experts survive this period on stockpiled foods will be working hard to genetically engineer better mushrooms and yeasts, while in the long term desperately searching for this new world's philosopher's stone: Fusion Power.
The Future of Power
If Fusion energy is developed, a small population of humans might live millennia is this cold night world.
Even if not, there is enough uranium for the current population of Earth to survive at its present level of consumption, for 200 years. If consumption of power becomes 1% of what it is now, nuclear power alone could keep humanity going for 2,000 years.
So, adding in coal and other fossil fuels, humanity should be able to survive with a reasonable population. Of course, without plants, the fear of carbon contaminating the atmosphere would become extreme. The fear would be much worse than the problem.
Even without a breathable atmosphere, it's quite practical to power a greenhouse and use it to recycle your air. This will be the main purpose of vegetables, to recycle air. The mushrooms are no good for this... they breathe oxygen too. So the more mushrooms you grow, the more vegetables you'll have to grow (like some kind of resource management sim!).
Mining and Scavenging
There will be a lot of scrambling in the early days to grab stuff from the stores (natural refrigeration) and steal supplies to survive. After people settle down into compounds, the main source of work will be scavengers braving the cold to get more supplies.
Eventually, when scavenging materials run short, we'll have to resume mining operations. In the extreme cold, it will be difficult to travel, prospect, and transport minerals. But the mining itself won't be too different. It's warm underground, and most of you will live underground anyway.
So, in the end, we might survive long enough to develop interstellar travel and adapt to this world - barring any unfortunate collisions with asteroids and meteors. Since the Earth is hurtling off into space all this time, this may even help us launch off towards the nearest star.