If the force of gravity at the Earth's surface suddenly became 2 times stronger, how would the environment and humans cope? I'm guessing a lot of flying creatures would suffer, but what would happen to us?
Every flying machine or creature would crash immediately and probably fatally.
Anyone with a weak heart, bones, or who is overweight would die fairly rapidly.
All structures such as bridges and large buildings would become potentially unsafe, depending on the safety margin with which they are constructed. Houses would probably stand but may need reinforcing, larger structures are at higher risk.
For details of the effects on humans see this answer: https://worldbuilding.stackexchange.com/a/171/49
As mentioned in the comments plants would struggle to support their weight, trees would collapse. Grain crops would not be able to stand. Many other plants would need artificial support such as tying them to frames to support them.
Depends on how it happens. There's three ways to double the gravity of Earth.
1. Increase the gravitational constant.
As Q said, "simple, change the Gravitational constant of the Universe". This is bad. Very bad. The Earth becomes more compact under its own increased weight and heats up. The Moon spirals into the Earth. The Earth and all the planets spiral into the Sun. Galaxies are disrupted... basically the whole Universe changes.
Let's not do that.
2. Double the radius of the Earth.
If you double the radius of the Earth, adding mass while keeping the density constant, it will have eight times the volume, and thus eight times the mass. But because the radius doubled, and because gravity decreases with the square of the distance, there will only be twice the gravity at the surface.
But this changes the surface which adds another change to deal with, so let's skip this one, too.
3. Double the density of the Earth.
If we double the mass while keeping the size the same we'll double the density, maybe by adding a lot more heavy elements below the surface. Because the size of the Earth is the same, and because gravity increases linearly with mass, gravity will double.
This is the best option because the surface remains the same... at first. Let's go with this one.
Our new, more dense Earth has problems, and not just for life.
For people, everyone will feel like they're giving themselves a piggy back ride. For those who can handle the strain, they will eventually gain in muscle and bone mass to compensate somewhat, but until then it will be an exhausting struggle to move or even breathe. Anyone too old, too young, or too sickly will likely be bedridden or die.
In addition to simply moving your greater bulk, your bodily processes become more difficult. Your lungs must expand and contract against greater mass, breathing becomes harder. Your heart must work harder against the increased gravity to get blood up to your brain, the increased blood pressure will put strain on your heart and blood vessels. Your lymphatic system, part of your immune system, is pumped by the action of your muscles and its fluids will tend to pool in your legs. There will be tremendous long term health consequences.
Most bridges and many tall buildings will fail. They are not built with a sufficient factor of safety to withstand doubling their weight. Airplanes will likely crash from either the increased weight, or the sudden atmospheric contraction (see below). Tall trees will either fall or quickly die from being unable to pump water to their tops, new trees will be much shorter. Many shorter plants won't have the structure to handle the increased weight and will be flattened.
Geological structures will collapse above, and more importantly, underground. Mines and other underground structures will collapse causing great sinkholes to open up. Cities are built on great systems of utility tunnels which will collapse. Others, built on poor soil and reclaimed land, will sink into the mire.
The increased mass means satellites in orbit will come crashing down. New launches will be harder, they have to with an escape velocity 1.4 times higher (escape velocity increases with the square root of gravity). But space will be closer! See below.
Double the mass means double the air pressure. Fortunately your body will reach equilibrium, so it won't make breathing harder. And you'll be receiving more oxygen, which is good news for all that extra work you need to do to move your new bulk around.
For anything moving quickly through the atmosphere, the increased air density will mean increased drag. Cars will be slower and less fuel efficient. Airplanes, on the other hand, will have greater lift and lower stall speeds.
A compressed atmosphere effects space launches... after all the satellites come crashing down. New launches will have to move through a denser atmosphere, which is bad, but the compressed atmosphere means the edge of space is lower. A compressed atmosphere means less drag on satellites which means they can stay aloft longer in closer orbits using less fuel... once they get there.
But human civilization probably won't survive on this new Earth...
The increased density has long term consequences for the Earth's habitability. All this extra gravitational energy means a hotter and more active Earth, this is bad for humanity.
The rapid compression of the atmosphere will cause a tremendous heating. This will generate great wind storms, pushing around more air mass, and strange weather patterns for years to come. It gets worse.
The increased density of heavy metals means a greater portion of radioactivity heating the core of the Earth. This means more seismic activity, more volcanoes, and more earthquakes in the long term. And it gets worse.
The increased gravity will cause the Earth itself to settle and compress adding even more heat. As the Earth shrinks, even a little bit, the surface will buckle causing great upheavals; new mountain ranges and volcanic eruptions will appear. All this mass shifting around will cause great earthquakes.
In the end, we're left with a slightly smaller Earth which is hotter and more geologically active. This means an unstable surface which is hostile to human civilization, much of which just collapsed under its own weight anyway.
Life on Earth is in trouble. Evolution has built on the assumption that gravity is a constant. Billions of tiny, critical biological processes have been built with that assumption. Now that it's suddenly changed, most complex life will die. The Earth will experience a great extinction event as it does every 100 million years or so. This will open great gaps in the biological niches, gaps which have not been open since multi-cellular life began, which will be filled by new forms of life. A few million years after this event, life on Earth will look very, very different.