Question: how does one go about inquiring climatological information?

The ways that you can look at changes in past and present climate are very similar.
Although you might not think it, clues from different types of micro- organisms (or microfossils if you are looking at past climate) can help us to unravel changes in climate. Microfossils are basically any fossils which have to be seen using a microscope, and which are more than 10, 000 years in age. Examples of these are pollen (powder produced by plants, these are not micro organisms they are just microscopic in size), foraminifera, coccolithophores (both single celled zooplankton made from calcium carbonate) and diatoms (single celled microscopic plants made from silica). These gorgeous little jewels in our ocean all have big climate stories to tell us. They preserve chemical signatures which can in turn be used to work out different palaeoenvironmental and palaeoceanographic conditions. For example, by studying the ratio of the concentrations of magnesium to calcium (Mg/Ca) in foraminiferal calcite we can work out past sea surface temperatures using various mathematical equations, and by studying neodymium isotope signatures (εNd) we can work out past changes in ocean circulation. Aside from looking at chemical changes, another way we can look at past climate is through assemblage changes i.e. changes in what species of creature were around and when. Some species like warmer climates and some prefer colder climates, and so by looking at which species are present are various points in time we can work out what the climate was like. To make this type of study more precise it really should be put together with chemical data. Climate modellers take all this data and put it into supercomputers in order to model both past and future climatic conditions.
One way of finding microfossils is through looking at ocean sediment cores i.e. giant, metre long ‘tubes’ of what is essentially, mud. These are wonderful archives of climate change and of how the Earth has changed in general. Sediment cores are normally taken from the ocean floor, and cores have been taken from all over the world. The International Ocean Discovery Program (IODP) is responsible for co-ordinating the research cruises that go out to sea to get them. However, sometimes cores are taken onshore. One such example of this is the Tanzanian Drilling Project which took place from 2002 – 2009. The sediment retrieved from this project is very special since it contains extremely well preserved foraminifera from the Eocene (56 million years – 33.9 million years ago) and Oligocene (33.9 million years – 23 million years ago). This is because the sediment was full of clay, meaning that rainwater was not able to trickle through to dissolve the fossils and reset their chemistry.
Ice cores are also brilliant archives of recent climate change. These record polar temperatures and atmospheric conditions. One of the longest ice core records stretches back ~600, 000 years, and is found at EPICA Dome C in Antarctica. Any type of dust (e.g. pollen, volcanic ash, sand grains) can get transported around the world, and so these are found in ice cores too. Evidence of rapid climate change can be seen in ice cores from Greenland too – these took place during the last glacial period (110, 000 – 12, 000 years ago) during the Pleistocene (2.8 million years – 11, 000 years ago).
Clues from (fossil) tree rings, (fossil) corals, speleothems (stalagmites and stalagtites) and rocks can also tell us what the Earths’ climate is like and used to be like too.

Concentrations of CO2 are recorded at the Mauna Loa Observatory in Hawaii daily, and can also be found on http://keelingcurve.ucsd.edu/