Below are some resources that I have found helpful in learning some of the mathematical methods used by economists and other social scientists.
A good way to learn some basics quickly is by reading the mathematical appendices of standard textbooks. The appendices to Varian’s Microeconomic Analysis and Mas Collel, Whinston and Green’s Microeconomic Theory outline the tools essential for economic theory, especially constrained optimization. The appendices to Greene’s Econometric Analysis provide a crash course in linear algebra, probability and statistics.
Once you know the basics, you’ll need a few good, general references to fill in the gaps. Mathematics for Economists by Simon and Blume covers the fundamentals of rigorous linear algebra, advanced calculus, optimization, and a few other topics. Mathematical Methods and Models for Economists by de la Fuente covers similar topics, and some more advanced material, with a more formal focus, although it is also very readable.
For probability and statistics, Hansen’s Probability and Statistics for Economists is a great reference that is geared towards econometrics. For a concise introduction, I recommend Gallant’s An Introduction to Econometric Theory (which also sketches the basics of measure-theoretic probability) and Wasserman’s All of Statistics (which includes brief introductions to some advanced topics). Two standard statistics texts are Mathematical Statistics by Hogg, Craig and McKean and Statistical Inference by Casella and Berger (the former being a bit easier and the latter being a bit more detailed).
I also recommend learning some basic real analysis, because it is directly useful, sharpens your mathematical abilities, and prepares you for more advanced material. There are many texts on analysis at this level (some of which is also covered by Simon and Blume and de la Fuente). For a concise introduction, I like Rosenlicht’s Introduction to Analysis, which is an inexpensive Dover reprint. The de facto standard reference is Rudin’s Principles of Mathematical Analysis, although, since it is more advanced and somewhat terse, it may be better as a second text or as a reference. For more discussion and motivation, as well as an introduction to some more advanced concepts, check out Carother’s well-written Real Analysis.
Knowledge at the level of the resources above is a good foundation for most applications. However, if you find yourself interested in even more technical literature, I have found all of the following to be (relatively) accessible introductions to and references for more advanced material.
Recursive Methods in Economic Dynamics by Stokey and Lucas with Prescott is about dynamic models, but it contains a good deal of mathematical background, including a good overview of measure theory. Rosenthal’s A First Look at Rigorous Probability Theory is a more complete, but still concise and readable, introduction to advanced probability. Luenberger’s Optimization by Vector Space Methods is a very readable introduction to functional analysis with applications to optimization. Two introductory texts on more advanced topics in real (and functional) analysis are Introductory Real Analysis by Kolmogorov and Fomin (another Dover classic, although a bit old-fashioned in places) and Axler’s Measure, Integration & Real Analysis (also see Axler’s intermediate-level linear algebra text, Linear Algebra Done Right).