I'm looking for a good book that gives an (introductory) overview about the existing economic theories and the different schools of thought (optimally with focus to those that affect governments and institutions). Presentations that use mathematical formalisms and that give a little bit of historical background to the respective theory are very welcome. I don't have any economic education, so I am hoping that there exist some "standard works" that compare the different theories and schools of thought.
Snowdon and Vane "Modern Macroeconomics: Its Origins, Development And Current State" is probably the best book I have read which gives a clear exposition of all the major macroeoconomic schools from the classicals to the New Keynesians. It highlights what the different schools have contributed to modern macroeoconomics and compares them to the other schools. They even include interviews with economists such as Robert Lucas Jr. and Milton Friedman to mention a few. I guess this is exactly the book you are looking for and to be honest I think every single person studying economics should read this book. It is not an introductionary economics book but I think most people should be able to read and understand it.
Because this question will ultimately produce subjective answers, and given the wise advice (given in the above comments) to avoid learning economics by studying "schools of thought," I decided to answer by telling you what books were assigned in my introductory economics class as an undergraduate.
In my first undergraduate class, our textbook was "Principles of Economics" by Mankiw (which is actually very nice to read even though it's a textbook). Along with this textbook we were required to read these two more popularly oriented books: "Naked Economics: Undressing the Dismal Science" (which has some fun discussion of economic issues) and "The Armchair Economist: Economics and Everyday Life" (which gives a great introduction to economics thinking).
These three books give a good beginner's overview of economics and economic thinking in general. I think this would be a nice way to start.
I would suggest the 2016's handbook "Handbook on the History of Economic Analysis, Volume II". It provides a comprehensive analysis of all schools of thought, from the Greek philosophers until today.
Volume III of the above handbook is also interesting, but it focuses on topics rather than schools of thought. For reference, Volume I contains a comprehensive, chronological list of economists, from William Petty to Paul Krugman.
Focusing on less general topics and more mathematically involved, you can combine Acemoglu's book on economic growth (a rather mainstream perspective) with the "Handbook of Alternative Theories of Economic Growth".
Also, Elgar put together a cool handbook of the most important papers on Capital theory, covering a whole range of perspectives.