I've reviewed a few books online for the MAA. When I learned undergraduate differential geometry with John Terrilla, we used O'Niell O'Neill and Do Carmo and both are very good indeed. O'Neill is a bit more complete,but complete, but be warned-the warned - the use of differential forms can be a little unnerving to undergraduates. That being said,there's said, there's probably no gentler place to learn about them. I DO do think it's important to study a modern version of classical DG first i.e. (i.e. curves and surfaces in R3, emphazing vector space properties) before going anywhere near forms or manifolds-linear manifolds - linear algebra should be automatic for any student learning differential geometry at ANY any level.
Of the textbooks mentioned here,I here:
I love Millman and Parker as well,although well, although it's not as complete as one would like. I'd love to see Dover put out a nice cheap paperback of it. Thorpe is ok,but OK, but doesn't excite me. His ; his notation gets unnecessarily dense. That being said,he said, he does emphasize linear algebra aspects and covers quite a few topics not found in the other texts.
Gray's mammoth tome is probably the single most complete source on classical DG:everything DG: everything is very clearly done with lots of fascinating computer drawn images and historical asides.But asides. But the incomprehensibly inserted program code is REALLY really distracting and breaks the flow and organization of the text-it text - it should be regulated relegated to software or online.For online. For that reason,I reason, I can't really recommend it as a class text,but text, but it definitely should be kept on reserve when teaching such a course.
Spivak and Frankel,although Frankel, although both wonderful texts,are texts, are really graduate level.Lastly,there's
Lastly, there are lots of free online resources for students now-the aformentioned now - the aforementioned lecture notes by Shifrin are outstanding, and we should enjoy them as long he makes them freely available before converting them to a real book. (Really looking forward to the finished product in a few years,though.........years,though...)