There's no arguing that John Lee's texts are excellent: the following are part of the series "Graduate Texts in Mathematics":
Each of the above links to Amazon, simply because you can preview the texts, e.g., the Table(s) of Contents, to see if any/all meet your needs. Each is also accompanied by credible "reviews", which may help you select the appropriate text(s) to meet your needs.
As you seem to be looking for a more elementary introduction to differential geometry:
You might want to check out the the course on Differential Geometry via MIT Open Course Ware, (Prof. Paul Seidel):
This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.
Other choices for Elementary Introductions:
- You might also want to look into O'Neill's Elementary Differential Geometry, perhaps a good choice to start off with.
"Written primarily for students who have completed the standard first courses in calculus and linear algebra, it provides an introduction to the geometry of curves and surfaces."
- Also look into the book with the same title: Elementary Differential Geometry, 2nd Ed (2010), [Springer Undergraduate Mathematics Series], this one authored by Andrew Pressley.
"Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout."