So the immediate answer to your question is that pre 1900's climate data comes from a variety of sources. The instrumental record actually goes back at least as far as 1850 (That is how far the Koch Brothers/ Berkeley data set goes for instance. ) predating that, when we look at time scales of millions of years we have to use various proxies Which ones are used and how precise they are depends how far back in time you go. The geologic time record goes back at least far as 500 million years
On twitter you rightly point out that extrapolation of future results based on past results is invalid. Even if we did have 4 billion years of perfect climate record, we would not be able to just fit a trend curve or line to that data and predict the next 100 years of earth's climate. You state on twitter that this past climate data is the basis for all of the rest of it. This seems to be a common misconception. There are many different aspects of climate science and global warming, and they are not all relying on past climate data as the basis of it. We do not simply extrapolate future temperature based on past data.
To use a simple analogy, suppose you have a sealed aquarium filled with air, with a heat lamp shining on it. You let the temperature stabilize, and then double the co2 concentration inside of it. We can predict what the temperature will rise to very accurately, without any past aquarium temp data at all, because we know the physics and chemistry of co2 and light very well. Models predicting the future surface temperature of the earth are using an approach like this, but with many more variables (methane, solar output, water vapor, etc etc) and with much more uncertainty. That extra uncertainty is why projections have ever increasing margins of uncertainty as the projection goes farther into the future. See for instance this projection The grey area is quite wide, and that is only a 95% confidence interval. So there is a great deal of uncertainty, but there does not seem to be any plausible scenario where it doesn't keep trending warmer, as long as we add co2 to the atmosphere at the rates we are, and the basis of this is physics, not past data.
What the past data does let us do is confirm some of our assumptions about our models. For instance we can use geologic data to confirm or deny a given theory about climate feedbacks (does warming from co2 tend to get offset by other negative feedbacks? past data suggests not, etc)
So the basis of the claim "The earth is getting warmer, and because of human activities" is really physics/chemistry. We can measure how much co2 has been put into our atmosphere from human activities directly. We know this because burned fossil fuels have a unique isotopic signautre. We know how much that co2 and methane should warm the earth based on physics. We have observed from satellites over the last 35 years that outgoing radiation has been reduced by the expected amount specifically in the co2 and methane spectra,which really removes any doubt as us having identified the correct cause of the observed warming. See Harries 2001