Tree rings can go back a couple of thousand years, which is > the 200 years you mention.
And biological markers within sedimentary rocks go back a lot further than that. TEX86 for example (membrane lipids of a marine organism) can provide a paleoclimate proxy to at least the Jurassic. Other examples include
Oxygen Isotopes and ratios of Magnesium and Calcium within fossil shell. They (and others) are reasons we do actually have a good idea of global temperatures during the Jurassic and Cretaceous. We also have a good idea of temperatures and CO2 levels in the Carboniferous (when a significant amount of the worlds coal was laid down, because of this). The latter shows a warmer earth with much greater levels of CO2. I believe the Jurassic and Cretaceous also had similar higher temperatures and CO2.
There are two issues with climate change.
One: The actual change - i.e. how hot it's getting and how much CO2 there is in the environment. Life is perfectly fine living in hotter locations with more CO2, can the current human however? Can we also cope with the more extreme weather events due to more energy in the atmosphere? More flooding, more hurricanes/tornadoes, more thunderstorms, rougher seas?
Two: The rate of change. This is the big one from an environmental point of view. Mass extinctions are usually caused by a sudden change in environmental factors (such as temperature and gases). The current rate of change is greater than many other times in history (excluding some of the mass extinction events). A rate of change too great means life struggles to adapt quickly enough, leading to the death of a significant proportion of life on earth. We know that most more complex life on earth cannot adapt quickly enough if the current rate of change continues, can humans? Can our crops? Can the fish that a significant proportion of us live off?
It's also worth pointing out that the current level of CO2 in our atmosphere is over 400ppm, which is literally off the chart (at least the chart you have posted showing the cyclic nature of our climate). Far greater than anything over the last 800,000 years. Perhaps a good indication that the current level isn't natural...? I would highly recommend looking into the PETM (Paleocene Eocene Thermal Maximum) to see what happens when large amounts of C02 are dumped into the atmosphere, and what happens if we reach the point of "no return" and the feedback loops start engaging, causing earth to vomit out huge amounts of Methane.
