Get this book: http://matrixeditions.com/UnifiedApproach4th.html and work through it if you want a rigorous treatment of MV calc and intro analysis.

Or this book: http://www.amazon.com/gp/product/0914098918/ref=pd_lpo_k2_dp... for regular calculus. Read them, work through the problems, get an answer key, email your solutions to professors asking them to look at them.

You will get a good introduction to analysis (and exposure to many other parts of math) from those two books.