Note: This year, the curriculum of Ma151 and Ma157 have been changed. In particular, the material here used to be covered in Ma151c. Thus those who took Ma151c last year will probably not be interested in this course, though they might well be interested in Ma157b.
The main text for this course will be
Gallot, Hulin, and Lafontaine, Riemannian Geometry, Springer-Verlag, Universitext, Second Edition, 1990 or later
Unfortunately, this book is out of print. I will get something set up where you can buy a photocopy of the text (not sure what this will cost as Springer hasn't gotten back to me about the copyright clearance fee). Alternatively, you could by a used copy online somewhere (e.g. here). With regard to this, the Second Edition was reprinted several times, and so appears with different publishing dates (1990, 1993, 1994, and 1996 have been observed in the wild). If the published date is 1987, that's the first edition, which you probably want to avoid. There will also be a copy of this text on reserve on the first floor of Millikan.
Initially, I'll start by covering some basic facts about smooth manifolds: vector fields, tensors, etc. (how much will depend on the classes' background). As the main text covers these topics in a fairly condensed fashion, you may also want to consult the following excellent source, which is on reserve on the first floor of Millikan:
Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press.
Another great book on Riemannian geometry is
Berger, A Panoramic View of Riemannian Geometry, Spinger, 2003.
This is not a textbook which carefully covers foundations of the field, but an 800 page attempt to survey all of modern Riemannian geometry. It is a great place to see what Riemannian geometry is all about, and also to get further intuition about basic concepts (there are several hundred figures and innumerable examples). I suggest you look through this book to find the topic for your paper, and then use the bibliography to get more detailed sources.
A copy of this book will also be on reserve in Millikan; unlike the others, I've set it up with a 3 day loan period to give you sufficient time to look through it. Also, Berge's book only costs $70 which is quite reasonable considering its bulk.
There is another point of view one can take on Riemannian geometry which deemphasizes the role of differentiability and focuses on more intrinsically metric-space notions. In particular, it is possible to talk about a general path metric space with curvature bounded above or below. This point of view is based on comparing geodesic triangles in your metric space with triangles in model geometries like the Euclidean plane and the round 2-sphere. This is called Comparison Geometry, and I sometimes find this point of view more appealing and geometric than the traditional one. The following book is a nice elementary account of this
Burago, Burago, and Ivanov. A Course in Metric Geometry. AMS Graduate Studies in Mathematics, Vol 33, 2001.
Finally, some other standard texts are