Primer on complex analysis and Riemann surfaces for undergraduate physics / theoretical physics majors Ref: The Road to Reality: a complete guide to the laws of the universe, (Vintage, 2005) by Roger Penrose [Chap. 7: Complex-number calculus and Chap. 8: Riemann surfaces and complex mappings]
I'm searching for an easily readable and understandable book (or resource of any kind; but preferably textbook with many worked-out problems and solutions and problem sets) to learn complex analysis and basics of Riemann surfaces - and applications to theoretical physics. (Particularly: material geared towards / written with undergraduate-level physics / theoretical physics students in view).
Any suggestions?
My math background: I have a working knowledge of single- and multivariable calculus, linear algebra, and differential equations; also some rudimentary real analysis.
 A: I don't think you'll be able to read a book on Riemann surfaces before you study complex analysis.  Once you've finished learning the rudiments of complex analysis, I recommend Rick Miranda's book "Algebraic Curves and Riemann Surfaces".  It probably is the gentlest introduction I know.
A: Theodore Gamelien's Complex Analysis requires only basic calculus, is very geometric and covers just about everything a mathematics undergraduate or graduate student needs to know about functions of a complex variable.It includes an excellent and very basic introduction to Riemann surfaces. For all those reasons,it's a must have for any mathematics or physics student at either level. 
A bit more sophisticated but equally wonderful is Singerman and Jones Complex Functions:A Geometric and Algebraic Approach-an incredibly rich and sophisticated second course for students who have already had an "epsilon-delta" type complex variables course and need to learn about the less analytic aspects of the subject. There's a terrific introduction to Riemann surfaces and meromorphic functions. 
There's LOTS of others-the already mentioned book by Needham is a treasure,if you're willing to overlook its sometimes loose approach.That should be more then enough to get you started!  
A: You may enjoy Needham's Visual Complex Analysis.
A: On the reference on Riemann surfaces: Forster's book "Lectures on Riemann Surfaces" is a great book that is easy to read and has many exercises (of course assuming knowledge of single variable complex analysis). 
A: As a beginning theoretical physics graduate student, incidentally I am these days going through the book "Introduction to Algebraic Curves" by Phillip Griffiths and it does seem to be a nice book for Riemann surfaces and such stuff. Also Jost's book on the same topic is great but focused in a different direction. For various things in complex analysis that I am once in a while getting stuck with, I seem to be able to pick them up from the book by Eliash Stein and Ramishakarchi. Somehow this combination of books does seem to work for me at least till now. 
As a physics student one has to enter mathematics "laterally".
Thats part of the challenge and thrill of physics whether or not the mathematical "purists" cringe about it :) 
