Books on differential geometry in the cases $n=2$ and $n=3$ I'm interested in learning the differential geometry of standard, "physical" space, that is $\mathbb R^2$ and $\mathbb R^3$. The sort of problems that were studied in the 18th and 19th century... curvature and so on. NJ Wildberger's lectures give an idea of the sort of content I'm looking for. I'm interested in books with a very down-to-earth, physical way of approaching things. Books that encourage thinking of a curve as a line through physical space, rather than a particular set of ordered triples, if you see what I mean.
To give a better idea of my needs, I've been struggling with multivariable calculus because all the definitions are in the context of $\mathbb R^n$ for arbitrary $n$, and I find it all very abstract and unmotivated. I feel I would get a better grip on the situation if I spent some time really deepening my appreciation for the concepts as they apply to familiar $2$ and $3$ dimensional geometry.
 A: I like A treatise on the differential geometry of curves and surfaces, by Luther Pfahler Eisenhart, edited in 1909, and available at Dover Phoenix. There are also Differential Geometry by Erwin Kreyszig and Lectures on Classical Differential Geometry by Dirk Jan Struik, both very valuable, and both reedited by Dover. They all follow a rather classic path to differential geometry, but I have the impression it's what you are looking for.
A: Personally, I like this one by Alfred Gray (originally). It covers most of the standard stuff, and also provides Mathematica code and a great many pictures (generated by Mathematica, of course).
Generally, if you search Amazon or a similar site for books with titles like "Differential Geometry of Curves and Surfaces" you'll find numerous options. Arguably the (modern) classic of the genre is this one by Do Carmo.
If you want something free/electronic (rather than paying for a paper book), then you might try Ted Shifrin's material.
I think you're absolutely correct to get a good grounding in $\mathbb{R}^2$ and $\mathbb{R}^3$ before venturing into higher dimensions.
Disclaimer: I use differential geometry (in $\mathbb{R}^2$ and $\mathbb{R}^3$) regularly in my work, but I've never taken a differential geometry class in my life. 
A: George B. Thomas Jr.'s Calculus and Analytic Geometry, while not a Differential Geometry book persay, has excellent discussions and examples that overlap considerably, at least in the early stages of study. I'm currently taking a Differential Geometry class and find myself referring frequently to sections from Thomas in order to fill in gaps in the discussion that I find in the course's assigned text (Differential Geometry and Its Applications, by J. Oprea). Perhaps as the course goes on I'll find less use for the Calculus book, but initially it's proving invaluable. This might apply especially for 2D and 3D.
