Book for Undergrad Differential Geometry I am soon going to start learning differential geometry on my own (I'm trying to learn the math behind General Relativity before I take it next year).  I got the sense that a good, standard 1st book on the subject was do Carmo's Differential Geometry of Curves and Surfaces and so that was the book I planned on reading.  However I just read this question on mathoverflow, and both answers to it suggested that the professor NOT teach a class from a book like do Carmo's because it doesn't cover differential forms.
Would you guys agree that I should find a book that introduces differential forms (and tensors?) given that I am an undergrad physics major who plans to study relativity theory?  If so, what books would you recommend?

I'm going to agree with Bryant in the mentioned link and recommend O'Neill's Elementary Differential Geometry. It is a gentle enough introduction to differential geometry, uses the common language and will prepare you for the usual problems in $\Bbb R^3$ while giving you a hint of what comes next. 
It may be profitably followed by his second book and/or John Lee's Introduction to Smooth Manifolds and Riemannian Manifolds.
