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?