First off, you don't state how much knowledge of maths and physics you _actually_ have beforehand, This makes answering the question an awful lot harder -- a 'college course in calculus' could be evaluating simple derivatives, or it could be some nasty vector calc and differential equations. In the order that they come into my head, you need to understand _intimately_ vector calculus (leading to Einstein notation -- play with it and become comfortable with it!), methods of solving partial differential equations, multivariate calculus, and how to properly play with differentials (i.e. proofs that start with statements like "df(x, y) = \partial f / \partial x dx + \partial f / \partial y dy"). You'll also need to properly understand matrix algebra, and ideally what tensors are (hint: generalisations of matricies that follow certain properties). You should be able to prove vector identities in Einstein notation, and be quite comfortable manipulating 'hardcore maths'. Honestly, just go away and play with maths until you understand it fully, you understand where it comes from, and you can use it without thinking about it at all. After that, try and become familiar with special relativity. This will be hard. Feynman explains everything very well in his lectures, but he doesn't list any problems: the best way to learn physics is to derive a true statement (like the lorentz contractions) and go away and shove it in all sorts of different situations (i.e. answer problems with it). The book by French & Taylor is commonly well-received; there are many different textbooks. Find a good set of problems, and answer them. Then, when you understand modern Special Relativity, get a large GR book -- there are many; Gravitation, or "General Relativity for Physicists" is a good one -- and read it. _Think_ about it, and answer the problems at the end of every chapter. If your book doesn't have questions at the end of each chapter, go away, and get one that does. Make sure you do them, and if you don't get something, find out why. If you can't find out why, ask someone who can.
Finally, a taught undergraduate level course in GR would be a fantastic introduction after a well-defined amount of knowledge has been acquired. The lecture notes from the course at my home institution can be found here.