Like you, I am a graduate student in the social sciences and I have a decent amount of stats (and calculus) classes under my belt. I tried several books, and the one that wins it BY FAR is "Doing Bayesian Data Analysis: A Tutorial with R and BUGS" by Kruschke (2010; see on Amazon).
The author, John Kruschke, also keeps an online blog about Bayesian statistics, that is a very nice companion to the book: http://doingbayesiandataanalysis.blogspot.com/
The book gives plenty of code, and all programs are available on the author's website. The book also comes with a set of applied exercises at the end of each chapter, and the full solutions are again available on the author's website. Very warmly recommended.