This is an introductory course to Differential Geometry, cross-listed as an advanced undergraduate course and graduate course. The main topics are the local and global theory of smooth curves and surfaces. Essential prerequisites are multi-variable calculus, linear algebra, and differential equations.
I particularly like Wolfgang Kuhnel's "Differential Geometry: Curves - Surfaces - Manifolds". The autor goes from curves to surfaces and from surfaces de Riemannian geometry in a very nice way. Even if you're not intersted in the manifold part it is a good book about curves and surfaces. The language is modern and the exposition of the subject very clear. It is better than Manfredo's book in my opinion.
I've reviewed a few books online for the MAA. When I learned undergraduate differential geometry with John Terrilla, we used O'Neill and Do Carmo and both are very good indeed. O'Neill is a bit more complete, but be warned - the use of differential forms can be a little unnerving to undergraduates. That being said, there's probably no gentler place to learn about them. I do think it's important to study a modern version of classical DG first (i.e. curves and surfaces in R3, emphazing vector space properties) before going anywhere near forms or manifolds - linear algebra should be automatic for any student learning differential geometry at any level.
Curves and surfaces by Montiel and Ros. A modern approach to the contents of Do Carmo's, but focusing on developing and using analytical methods, particularly integration. This book is actually used for an introductory course on the geometry of curves and surfaces at my home university (Granada). 2b1af7f3a8