I wanted to get this written down before I forget my points, and I figured I'd share. This is later on in my intended argument, and this will likely be refined, but it'll give an idea of where I'm going, in terms of trying to execute this in a honest scientific manner.
Of course, my intention is to back up my claims below with proper research, when I get to those parts.
Engage Brain Dump
We can look at the trends of CO2 and temperature.
We could look at every permutation of positive, neutral and negative trends for each, but since the trends for both are positive, it would be moot to examine the permutations of the others, since they aren't happening.
If we look at CO2, there's a few basic possibilities:
1) The CO2 rise is from humanity (even if we're simply nudging it up a little at a time)
2) The CO2 rise is not coming from human output.
#2 is not supported by any evidence. In fact, it would have two implications:
1) The human CO2 output is negated somehow (disappears, made inert, etc)
2) One or or more other sources are responsible for the rise
There's no indication that either of these are true. CO2 is a stubborn persistent gas, and outside of oceanic or soil absorption, the only other thing on this planet that effectively removes CO2 from the atmosphere is plant life - something that humanity has been extinguishing concurrently.
One may argue that volcanoes are a major output for CO2, which may be the case, however, they are also a major source of ash at the same time, which effectively drops the global temperature, as we've seen with many eruptions. If volcanoes were the source, we'd see an upward trend of CO2, and a downward trend of temperature. We don't see that in the recent CO2/Temperature increases. Volcanoes are thus ruled out.
Meteorite or comet impacts would have the same result as volcanoes, except with dust, but with little CO2 output. We haven't had any major events like that in quite awhile (we'd notice the equivalents to "nuclear winters")
Geologically speaking, if one ignores the CO2 levels over history, and focuses on the change in levels, the last 70-100 years are unprecedented in two major ways. First, that the rates are notably faster than "natural cycles" typically are (to a significant margin), and secondly, that the quick changes are typically downward for temperature - when meteors/comets impact or big volcanoes erupt. There's no precedent for sudden upward trends in temperature outside of global catastrophes that result in the planet's surface being molten.
Ignoring humanity's significant CO2 output is ignoring the elephant in the room. It's pure denial-ism.
If we look at the temperature change, and what might account for that outside of CO2 levels, there are a wider variety of possibilities:
The first is solar output. If we ignore the lab-demonstrable greenhouse effect attribute of CO2, it is possible for the planet to heat up without an increase in CO2 levels merely from an increase in solar output. If the solar output is consistent, or lower, that would refute this possibility that we were warming up merely to do the sun heating up.
It should be noted that we couldn't merely dismiss the greenhouse-gas nature of CO2, no matter whether the solar output is up, down, or neutral. It would be like going outside in the sun, throwing on a jacket, and asserting that the solar output just happened to increase at the same instant that you put on the jacket, especially when it's incontrovertibly demonstrable that jackets are a heat insulator. That's how silly the argument would have to be.
As it turns out, while the solar output did increase in the 1950s to 1960s, it's been mostly consistent since then, which on a basic level refutes the solar-output argument. What's even more damning is that in the past decade, when temperature records have been repeatedly broken year after year, with the last 7 or 8 hottest years on record being in the last decade, the solar output has been at a decline. Solar output is thus ruled out.
A quick look at the 2nd law of Thermodynamics shows that if we removed the sun from the equation, we'd have little to no incoming energy. The planet isn't going to spontaneously heat up unless untapped energy sources were released (nuclear, chemical, etc). That hasn't been happening. Instead, we have that elephant in the room - the extra jackets we're putting around our planet, which, like during the winter, are enough to keep you warm, even when one is getting less heat from the sun.
It's possible that something else has been absorbing sunlight that's not a greenhouse gas - and that's true - the oceans. As the polar ice caps have been melting, less of the icy surface has been reflecting away solar radiation, and more ocean is available to absorb it. This accelerates the temperature increase, especially at the poles. This is demonstrably confirmable It, however, is not the cause of all this, and we can determine that due to its latency. The caps haven't been frighteningly melting until much later than the CO2/Temperature increase trends. Oceanic absorption, at least as an initial cause, is ruled out.
Blanket dismissals via spouting off "Yes but natural cycles" says nothing. It'd be like arguing that since the water levels of a pond have been higher or lower in the past, that therefore our building an aqueduct, pumping a few thousand gallons of water per hour into the pond would have no effect on raising the water level. The long term natural trends of the pond water level has no relevancy to the demonstrably true mechanism that adding water to the pond now will raise the levels, based on a rudimentary understanding of the laws of physics. "Natural cycles" is thus ruled out.
What we've come to is that we have a demonstrably high human output of a lab-demonstrable greenhouse gas, and a demonstrably increasing CO2 trend with a demonstrably increasing temperature trend, with possible rebuttals having been refuted. We have a theory that accounts for most all the available data in the simplest fashion (application of Occam's razor), and is supported by the preponderance of evidence.
Withing science, this is about as "proved" as something can get.