List of problem books in undergraduate and graduate mathematics
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.
General Topology:
V. Arhangel'skii, Fundamentals of General Topology .
K. Rao, Topology.
Differentiable Manifolds:
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.
E. Pap, Complex Analysis through Examples and Exercises.
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.