I'll try to keep it brief.
1. CO2 is a greenhouse gas
John asks how we know that atmospheric CO2 traps heat:
We cannot measure directly how atmospheric CO2 traps heat. How would we? You can't put a thermometer in the Earth's mouth. We can't build a series of otherwise identical Earths and release different amounts of CO2 into their atmospheres. What direct measurement does he have in mind?
It's a physical property of CO2 as a gas, which we can measure directly. You take a vessel with CO2 in it, shine infrared radiation through it, and measure the absorption.
Our knowledge that CO2 in the atmosphere traps heat isn't dependent on models of the climate, or surface temperature measurements, or reconstructions of past temperature records. We can measure it directly.
Also: we have satellite measurements of infrared radiation leaving the earth, going back to 1970, and we can see that CO2 and other greenhouse gases in the atmosphere are trapping more and more thermal radiation. (We can see that there's a drop in the specific parts of the spectrum where CO2 absorbs radiation.)
So yes, CO2 is a greenhouse gas.
2. Sources of information
For readers who find any information provided by "alarmists" such as myself to be dubious, you may find Richard Lindzen more credible. He's not an "alarmist," but a prominent skeptic: according to the New York Times, "When he appears at conferences of the Heartland Institute, the primary American organization pushing climate change skepticism, he is greeted by thunderous applause."
Lindzen and Choi (2011) explain how greenhouse gases work:
The heart of the global warming issue is so-called greenhouse warming. This refers to the fact that the earth balances the heat received from the sun (mostly in the visible spectrum) by radiating in the infrared portion of the spectrum back to space. Gases that are relatively transparent to visible light but strongly absorbent in the infrared (greenhouse gases) interfere with the cooling of the planet, forcing it to become warmer in order to emit sufficient infrared radiation to balance the net incoming sunlight (Lindzen, 1999). By net incoming sunlight, we mean that portion of the sun's radiation that is not reflected back to space by clouds, aerosols and the earth's surface. CO2, a relatively minor greenhouse gas, has increased significantly since the beginning of the industrial age from about 280 ppmv to about 390 ppmv, presumably due mostly to man's emissions.
It is generally accepted that in the absence of feedback, a doubling of CO2 will cause a forcing of [approximately] 3.7 Wm-2 and will increase the temperature by [approximately] 1.1 K (Hartmann, 1994; Schwartz, 2007).
Rephrasing: doubling atmospheric CO2 is equivalent to increasing the solar energy reaching the earth's surface by 3.7 watts per square metre. Without calculating feedbacks (e.g. a warmer atmosphere holds more water vapor, and water vapor is also a greenhouse gas), we would see a temperature increase of 1.1 degrees C.
3. Next steps in the argument
So far the line of reasoning looks like this:
Atmospheric CO2 traps heat. We can measure this directly.
We can directly measure the level of CO2 in the atmosphere and see that it's steadily increasing.
Thus we can expect the amount of heat trapped by the atmosphere to increase over time.
But of course the next question is: How much of an effect will it have on the climate? What's the magnitude? What makes Lindzen a skeptic is that he thinks the resulting change in temperature won't be enough to be dangerous. Based on his calculations (presented in Lindzen and Choi 2011), he thinks doubling CO2 will result in a temperature increase of 0.6 or 0.8 degrees C.
If we accept the line of reasoning so far, the next argument is:
Yes, we can expect increasing CO2 to trap more heat.
But it won't cause enough of an increase in temperature to be dangerous. Perhaps it'll be tiny compared to natural variation.
Before continuing, I'll pause to give John some time to respond. After that, I'll go into what we know about the climate (e.g. feedbacks) and the expected increase in temperature from increasing CO2. Would we expect the increase in temperature to be dangerous, or not? As John says in his post, there's lots of issues here to discuss: complexity of the climate, sources of natural variation such as solar activity, accuracy of temperature measurements, what we know about past climate change, dealing with feedbacks, causes vs. effects.
Thanks to John for taking the time to respond to my initial comment!