Is it better to shoot with a higher ISO, or use lower ISO and raise the exposure in post-processing?

If you have a tripod and you can, use low ISO. If you cannot use a tripod, increase the ISO because grainy images are better than shaked ones. A very dark picture cannot be recovered (as a burned one) once it gets "full dark areas" because there is nothing to recover (no details). So, if you can see things through your viewfinder and after taking the photo you still have details, it's OK. Also, check the histogram.

To correct your statement a little,

"A higher ISO will produce a more grainy image at standard exposure."

The ISO used has no impact on the light entering the camera or the sensitivity of the sensor to light. A certain amount of signal must be registered to mute out the noise that is inherent in the system. At a high ISO, the sensor processes a smaller amount of light as more light and thus develops an image faster, but has more noise.

That same image, taken with a lower ISO, but at the same speed will still have the same amount of noise, but as Matt Grum points out, the digital conversion will likely amplify the noise and thus make a worse overall image as the levels going in will be so low when still in analog form.

The reason that there is less noise at standard exposure is that the additional time that light is collected by the sensor results in more signal compared to the relatively constant amount of noise.

Basically you get noise from both power fluctuations in the camera and in noise from the image sensor if you use a lower ISO and don't do a standard exposure.

Formally, it is better to go with higher ISO, as long as higher ISO is implemented the way it is supposed to be implemented: as higher gain setting for the internal analog signal amplifier in the camera, i.e. signal amplifier that works in the signal before it gets converted to discrete digital form.

From the purely mathematical point of view, internal amplifier also "multiplies" the signal, just like "Exposure" slider in some post-processing software does. However, the multiplication is performed on the analog side of the processing flow, before the signal is converted to digital form. This precludes any rounding errors that are invariably introduced when the multiplication (or virtually any other image value transformation) is performed on the discrete digitized values.

Of course, in order for all this to work, your camera must implement the ISO setting specifically by adjusting the gain of the internal analog amplifier. Cheaper digital cameras might implement "fake" ISO settings by keeping the amplifier gain constant and multiplying the image data after it was converted to digital form. Such implementation will offer absolutely no meaningful benefits over what you can do in post-processing. It is precisely the same thing as the "Exposure" slider in post-processing software.

Meanwhile, many modern SLR cameras implement majority of their ISO settings by adjusting the amplifier gain, while "simulating" the most extreme ISO settings (the highest and/or the lowest) by digital multiplication. There's no point in using such simulated ISO values for the reasons mentioned above. Some cameras might implement "standard" ISO values (lie 100, 200, 400, 800 etc) by adjusting the amplifier gain, while simulating fractional ISO values by digital multiplication from the nearest "standard" ISO value. Again, there's no point in using such simulated ISO values.

If the iso 3200 is not overexposed then the iso 100 is very dark, not just quite dark. In 8bit range it would go between 0 and 15 pixel values, so very noisy, and the iso 3200 would be quite fine, except in dark regions.

These images are ISOs 200,400,800,1600 from left to right on an old Canon 400D where high iso was known to be bad. But still the ISO 1600 is the best.

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Indeed, the best way to lower noise is by adding light to teh sensor, through settings or by adding light to the scene. These operations come with a cost: adding light changes the scene we want to capture, adding light through exposure time risks motion blur which can be worse than the noise, and opening the iris (if possible) decrease the DOF and makes the lens soft, and maybe even add some fringing.

Here I added flash, iso 200-800 (1600 would overexpose):

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Here we see that now there is enough light to make a low noise image in all cases - especially notice that going up in iso still doesn't make it noisier. Lack of light makes it noisy, not high iso.

In conclusion, light > iso, but higher iso != your enemy; it is your friend when the light fails you.

It depends on the camera. Increasing the ISO boosts the signal from the sensor, and this is different than brightening the image in post. High ISO can be thought of as a hardware boost while brightening in post can be thought of as a software boost.

Increasing ISO might result in less noise, it might not, compared to post processing brightening. Most cameras boost the signal only up to a point (400 to 1000 ISO or so) and achieve effective ISO increases on top of that by software manipulation. Again, at what point this happens varies by camera model.

You'd have to test your specific camera model to know how the effects of high ISO and PP brightening differ.

In any case, you want a good exposure, that is you want raw data with the most useful information in it, and the expose to the right technique is usually a good way to achieve this.

You are talking about a digital camera here. If at equal exposure, quality could be improved by lowering the ISO value and digitally amplifying the signal, why would the camera not do it by itself? Where would be the point in artificially making the image worse than it can be?

With film, you could argue that similar techniques could make some sense since exposure and lab processing are two different stages controlled by different actors, and if you had control over both actors rather than just one, you could improve results in a manner that you couldn't by only controlling exposure.

But the digital photograph produced in the camera is completely under control of the camera. If underexposing and then amplifying the result digitally would be the best way forward, that is how ISO would be implemented (in fact, "ISO invariant" sensors like newer Sony Exmor sensors do it in that manner; but with numerous other/older/smaller sensors the circuitry in the analog domain can do a better job when it anticipates lower exposure).