Big list: Free textbooks and resources
Ravi Vakil's The Rising Sea: Foundations of Algebraic Geometry is an excellent textbook on (you've guessed it) foundations of algebraic geometry, very nicely mixing rigorous proofs, intuitive explanations and illustrations. It is available for free on author's website, and the newest version (along with all other versions) is always available here:
http://math.stanford.edu/~vakil/216blog/index.html
There is also Fulton's book Algebraic Curves: An Introduction to Algebraic Geometry is another nice book whose latest edition has been published for free by the author. It gives an elementary introduction to the theory of algebraic curves, up to desingularization and Riemann-Roch theorem.
http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
Both books require some familiarity with abstract algebra, but assume no prior contact with algebraic geometry.
Project Gutenberg is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks". It was founded in 1971 by American writer Michael S. Hart and is the oldest digital library. Most of the items in its collection are the full texts of public domain books. The project tries to make these as free as possible, in long-lasting, open formats that can be used on almost any computer. As of 23 June 2018, Project Gutenberg reached 57,000 items in its collection of free eBooks.
-Wikipedia
In a similar vein, https://github.com/rossant/awesome-math is another compilation of good math resources, including lecture notes.
http://builds.openlogicproject.org/ is an excellent resource on logic, beginner to advanced.
http://abstract.ups.edu/aata/index.html is something I have not read personally, but am told that it is a good resource for abstract algebra.
For some introductory number theory topics, I would recommend Stein (2017), which covers areas from congruences to cryptography to continued fractions. Only some basic knowledge of properties of numbers is needed, and the book also drops in SAGE code for examples here and there.
Likewise, the textbook on complex analysis by Cain (1999) is elementary but shorter; major topics include calculus and series. Knowledge of these topics on the real numbers is needed, as some parts of the book are quite involved. Both these books have exercises at the end of each chapter which vary in difficulty.
For a large list of free textbooks (on many areas on mathematics) kindly offered by their authors, visit OpenCulture.
References:
Stein, W. (2017). Elementary Number Theory: Primes, Congruences, and Secrets. Available from: https://wstein.org/ent/ent.pdf.
Cain, G. (1999). Complex Analysis. Available from: http://people.math.gatech.edu/~cain/winter99/complex.html.