Here’s a nice overview of causal inference by Scott Cunningham. Yes, you get an entire book as a free download, and it’s got you covered from probability to Pearl’s directed acyclical graphs, from instrumental variables to synthetic control. It comes across quite friendly, but has enough econometrics to scare many off. I quite enjoyed the historical bits thrown in here and there to explain where the methods came from.
Kosuke Imai has recently published a great introduction: Quantitative Social Science: An Introduction. Finally a
stats data analysis book that has arrived in the present! Yes, we can get away with very little mathematics and still do quantitative analysis. Yes, examples from published work are much more interesting than constructed toy examples. Yes, R can be accessible. Yes, we can talk about causality, measurement, and prediction (even Bayes) before getting to hypothesis testing. Yes, we can work with text and spatial data.