I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, topology etc. The books should contain solution to many of the problems if not all, something similar to Problems in Algebraic Number Theory by M. Ram Murty, Jody Esmonde. Also, any text book contain solutions to many of exercises will also do.
The list of books that I would recommend to you,
Problems and solutions in mathematics, This book contains a number of questions and solutions on topics such as Group and Galois Theory.
P. Gadea and J. Masqué, Analysis and Algebra on Differentiable Manifolds.
S. Morita, Geometry of Differential Forms.
Mathematical, Real and Complex Analysis:
D. Aliprantis and O. Burkinshaw, Problems in Real Analysis.
W. Kaczor, Problems in mathematical analysis.
R. Shakarchi, Problems and solutions for Complex analysis.
B.Makarov, Selected problems in real analysis.
and Schaum's Outline of Theory and Problems are a series of supplementary texts as problems and solutions.
You can begin with the Red and Green books of mathematical problems,both available through Dover. Then there's a ton of good ones,mainly on analysis and combinatorics. A classic is the 2 volume Problems in Analysis by Polya and Szego. Very difficult,but well worth the effort. Before that, you might want to cut your teeth on the wonderful The Cauchy-Schwarz Master Class by J. Micheal Steele, to learn many of the basic inequalities that will be needed there.
After that,the sky's the limit. From T.S. Blyth's 5 volume problem course on abstract algebra to Combinatorial Problems and Exercises by Laszlo Lovasz to the 2 volume Problems in Mathematical Analysis by W. J. Kaczor and M. T. Nowak.
Get pen and paper and get started!
Springer has an entire series titled Problem Books in Mathematics. The two by Lam are well known and have solutions. I don't know if all the books do.
A Collection of Problems on Complex Analysis by Volkovyskii et al. (Dover) takes problems from several well-known texts and furnishes answers for many. I am finding it helpful--problems range from very basic to pretty advanced. It may not be part of the Problem Book series Jim mentions.
And it's inexpensive.
I mentioned these in a comment, but they might as well be added to the collection in the answers. Graham, Knuth, & Patashnik, Concrete Mathematics, Miklós Bóna, Introduction to Enumerative Combinatorics, and Herbert S. Wilf, generatingfunctionology, all include solutions to a great many of the problems.
There is a great list maintained online on http://www.mathpropress.com/mathBooks/. It contains huge list of problem books in most branches of mathematics, even those especially for physicists.
Here is a list of my favorite math (and mathematical physics) problem books.
Lectures, Problems and Solutions for Ordinary Differential Equations by Yuefan Deng
Problems and Worked Solutions in Vector Calculus by L.R.Shorter
Problems and Solutions in Introductory and Advanced Matrix Calculus: Second Edition by Willi-Hans Steeb & Yorick Hardy
Introduction to the theory of Statistics by Mood, Graybill (doesn't come with solutions but a solutions manual is floating out there on the web)
Baby Rudin plus A Complete Solution Guide to Principles of Mathematical Analysis by Yu or the other (partial) solutions manuals out there on the web.
There are many many more, this is just a random sampling