For me it tends to be easier to take a historical view of a scientific topic, exploring what was known at a given time, and particularly focusing on controversies. AGW is widely accepted now, but has been extremely controversial, and was actually entirely discredited for some fifty years after the first published paper on the topic. The reasons for this should probably be extremely popular among skeptics.
The Discovery of Global Warming [0] is a hyperlinked and well-cited ebook which describes the climate research of the last century or so.
[0] https://history.aip.org/climate/index.htm
In the 19th Century the prevailing view of climate was that it was static or cyclical, with cold years balancing out warm ones. This began to be challenged by growing evidence for past ice ages, and various theories of climate change proposed mechanisms by which these might occur. At around the same time, people began playing around with carbon dioxide, carbonated water, and carbonic acid, and noticed that many human activities produced large amounts of CO2. In the mid-1860s Tyndall measured the heat characteristics of various atmospheric gases (you can probably reproduce his experiments relatively easily, if you like). A few decades later in 1896 Arrhenius suggested that halving the amount of CO2 in the atmosphere could produce temperatures cold enough for an Ice Age, but he also provided figures for a doubling of atmospheric carbon. His value for climate sensitivity is a bit on the high side of today's range, but still agrees pretty well despite his unsophisticated climate model.
However, Arrhenius was refuted in 1901 by Knut Ångström, who pointed out that 1) the absorption spectrum of water and CO2 overlap, and the atmosphere is essentially completely saturated with H2O, so carbon dioxide probably isn't having any additional effect, 2) the atmosphere is completely opaque to CO2 at lower concentrations than currently exist, so additional CO2 should have no effect, and 3) that the oceans are an unimaginably capacious carbon sink, and can absorb all the carbon that humans could even think about liberating, and then some. The CO2 theory of climate change was mostly forgotten for about half a century.
In that time, we began to explore the upper atmosphere, as well as the circulation of the oceans. Other experiments shed doubt on the cyclical nature of climate, pointing out that (e.g.) small changes in albedo could reflect sunlight, leading to cooler temperatures, and so on, in a self-reinforcing cycle. The idea of a cyclical climate took a long time to die, and one of my private amusements was reading through a 1950 textbook on atmospheric science. It described the climatic zones of the world as if the annual rainfalls and prevailing winds were graven in stone, and explicitly assigned a small role to CO2. By that time, however, the scientific view was already beginning to change.
In 1949 Callendar published an article titled Can Carbon Dioxide Influence Climate?, which laid out a renewed case for the importance of CO2. The absorption spectra of water vapor and carbon dioxide do not completely overlap, he argued, and although in theory the oceans can absorb a nearly-infinite amount of CO2, the rate of mixing of the upper and lower oceans is very low, and so the oceans provide much less of a buffer in the short term. He also pointed out that the stratosphere was almost devoid of H2O, and that small increases in the composition of the outer atmosphere could have a disproportionately large effect. More generally we can say that increasing the partial pressure of CO2 increases the extent of the CO2-rich layer, raising the effective top-of-atmosphere.
There were a number of unknowns at this point. It was not known for sure whether solar output was constant, or whether the global concentration of carbon was increasing, or whether the influence of carbon dioxide would be outstripped by the influence of particulates or other polluting gases. Solar observations since then suggest that solar output is constant to within .1 percent. The potential cooling effect of particulates and aerosols was the topic of active debate until the mid-1970s, and I'm told that some periodicals published lurid extrapolations of this research. Measuring the global concentration of CO2 was something of a challenge, but it was eventually met in 1958-61 by one Charles Keeling, who established a global baseline for CO2 concentrations, showed a large seasonal variation in the same, and finally showed that the levels of CO2 in the atmosphere were indeed increasing as predicted.
Since then we have seen steadily rising atmospheric concentrations and global temperatures. Better atmospheric modeling and stricter pollution controls reduced the potential threat of cooling. Another potential avenue of escape was offered by the complexities of H2O interactions. By itself, CO2 is not all that much of a concern. The no-feedback forcing per doubling is calculated at ~3.7 W/m^2, which is generally held to be about equivalent to 1 degree of warming per doubling. And if that were all we could expect, we probably would be talking about ocean acidification instead of global warming. However, water vapor is a much stronger greenhouse gas than CO2, and there's quite a bit of water lying around most everywhere here, and the atmosphere can hold exponentially more water the warmer it becomes, so a naive calculation would suggest an unbounded positive feedback loop. Fortunately this is not observed. The interactions of water in its various states are quite complex, so at this point we essentially had to leave the laboratory and try to study the Earth as a system.
That part is hard. Warmer tropical currents could shut down the Gulf Stream, which would probably result in Europe freezing solid, and ironically provide the impetus for a new Ice Age. (opinion) The strongest skeptical argument presented recently would probably be Lindzen's Iris hypothesis, which suggests that increased cloud coverage could offset rising temperatures. Unfortunately the balance of evidence suggests otherwise. "Something lurking in the H2O feedback" is at this point about all that would save us, and any potential mitigating effect would generally have to be both large in magnitude, to counteract the H2O feedback, and also presumably small enough not to have been noticed. Without getting spectacularly hand-wavy with physics, one might also posit some unknown interaction in the upper atmosphere which would conveniently transfer large amounts of heat into space. The oceans are pretty much ruled out, the optical properties of CO2 are beyond dispute, and hoping for an exception to the laws of thermodynamics is probably a tad optimistic.
The linked ebook should provide adequate citations for all of the above, and the research papers should all be freely available online. Do please respond if you have any further questions, or if you would like any assistance in finding citations. Also, while I have read a fair amount on this subject, I am a layman, not a climate scientist, and the above being somewhat extemporaneous, I would also be appreciative of any chance to correct any mischaracterizations.