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I recently went through Griffiths' Quantum Mechanics text and there is a chapter called the Adiabatic Theorem that includes Berry phase and the Aharonov-Bohm effect.
As I found them very interesting, I would appreciate if anyone could provide me with some good sources(books, internet links etc) to learn more about them.
Please keep in mind that I am only an undergraduate student, so difficult graduate stuff might not be of very use to me.
I suggest the American Journal of Physics as a good reference for undergraduates. The readership is Physics teachers so usually covers the topics that are of interest to students and the parts of a research subject that are of most interest to students. Plus it tries to make thing accessible to people that aren't in the specific subfield (in case they are teaching that subject) so they are generally accessible.
The article "The adiabatic theorem and Berry's phase" by Holstein in American Journal of Physics 57 (12) 1989 pp:1079-1084 DOI:10.1119/1.15793 is fairly introductory, rather detailed, and explains what it is doing. Though it does tend to refer to popular things it thinks you might know, things that in your case you might now know. However you can supplement it with the next paper.
The paper "Comment on the Adiabatic Condition" by A.C. Aguiar Pinto, M.C. Nemes, J.G. Peixoto de Faria, M.T. Thomaz in American Journal of Physics 68 (10) 2000 pp:955-958 DOI:10.1119/1.1285944 goes into detail of solving the actual dynamics first, then making approximations later so you know what approximations are made, as well as the why and how.