Introduction
Start with Poonen's "Computational aspects of curves of genus at least 2." This is a great survey article with which to get everyone warmed up.
Once you are primed you can go to the source, Coleman's "Effective Chabauty". It's a fairly readable paper if you assume that the padic integration results work as quoted. Note: think about what a residue class is  how does it relate to choosing a model for the curve?
The next two are optional, but you might want to read the article by Grant and Chabauty's original note at this point. Grant because it is easy and Chabauty for the historical interest. Also, Chabauty's note is a bit vague on the padic analytic foundations, which serves as a motivation to look at these foundations more closely. Did Chabauty actually prove anything? (Chabauty's note is a bit hard to find, email me if you want me to send you a copy.)
There are at least four approaches to the padic analysis.

Coleman, "Torsion points..."

Flynn, "A flexible method ..."

McCallum, "The arithmetic of Fermat curves"

Wetherell, Ph.D. Dissertation, Chapter 1 (see attached)
I will take my dissertation as the required text and the other three as optional, but I recommend you at least skim through all four of these papers. It is a good idea to get a sense of the various different approaches. We could have a good discussion of how these approaches relate to each other.
That maybe does it for an introduction. Then we divide into two groups.
Computational ColemanChabauty
This group will go back and read my dissertation and Flynn's "Flexible method" more thoroughly. Flynn, Poonen, Schaefer is also very good. Then we extract the necessary bits from Flynn and Wetherell (not yet in press, I will send it in email). I talk you through putting the pieces together for a specific, challenging problem.
(1zeta_n)descents on cyclic ncovers of P^1
The definitive paper here will be McCallum's Inventiones article. Other useful references are Poonen $amp; Schaefer and Schaefer's "Computing ... using functions on the curve". Bill will be putting together the basic description for this project.
General Citations Poonen B
Computational aspects of curves of genus at least 2
LECTURE NOTES IN COMPUTER SCIENCE 1996, Vol 1122, pp 283306
Coleman RF
Effective Chabauty
DUKE MATHEMATICAL JOURNAL 1985, Vol 52, Iss 3, pp 765770
Grant D (optional)
A curve for which Coleman effective Chabauty bound is sharp
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 1994, Vol 122, Iss 1, pp
317319
Coleman RF (optional)
Torsion points on curves and padic abelianintegrals
ANNALS OF MATHEMATICS 1985, Vol 121, Iss 1, pp 111168
Flynn EV (optional)
A flexible method for applying Chabauty's Theorem
COMPOSITIO MATHEMATICA 1997, Vol 105, Iss 1, pp 7994
McCallum WG (recommended but optional)
The arithmetic of Fermatcurves
MATHEMATISCHE ANNALEN 1992, Vol 294, Iss 3, pp 503511
Citations for Computational ColemanChabauty Flynn EV (again, required here)
A flexible method for applying Chabauty's Theorem
COMPOSITIO MATHEMATICA 1997, Vol 105, Iss 1, pp 7994
Flynn EV; Poonen B; Schaefer EF
Cycles of quadratic polynomials and rational points on a genus2 curve
DUKE MATHEMATICAL JOURNAL 1997, Vol 90, Iss 3, pp 435463
Citations for (1zeta_n)descents on cyclic ncovers of P^1 McCallum WG
On the ShafarevichTate group of the jacobian of a quotient of the Fermat
curve
INVENTIONES MATHEMATICAE 1988, Vol 93, pp 637666
Poonen B; Schaefer EF (optional)
Explicit descent for Jacobians of cyclic covers of the projective line
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK 1997, Vol 488, pp 141188
Schaefer EF (optional)
Computing a Selmer group of a Jacobian using functions on the curve
MATHEMATISCHE ANNALEN 1998, Vol 310, Iss 3, pp 447471
