Just as you mention it, I strongly recommmend the new edition of Tu – “An Introduction to Manifolds” since it is accessible but also very well-organized and motivated and basically starts up from multivariable calculus and ends up with cohomology of manifolds (it is very useful for example to get the needed background to follow his other more advanced and topologically focused text Bott/Tu – “Differential Forms in Algebraic Topology”). Moreover it includes hints and solutions to many problems!.
A little bit more advanced and dealing extensively with differential geometry of manifolds is the book by Jeffrey Lee – “Manifolds and Differential Geometry” (do not confuse it with the other books by John M. Lee which are also nice but too many and too long to cover the same material for my tastes). You can use it as a complement to Tu’s or as a second reading. It is much more complete since it deals with all the stuff in Tu’s but includes a lot more like vector bundles and connections, Riemannian geometry, etc.
In the same spirit of the previous book but a little better in my opinion, and even more complete, is the title by Nicolaescu – “Lectures on the Geometry of Manifolds“. Its table of contents is amazing in scope dealing with some advanced topics most other introductory books avoid like classical integral geometry, characteristic classes and pseudodifferential operators. It supposedly builds everything up just from a background in linear algebra and advanced multivariable calculus. It may seem a little bit advanced at first, but it is the best book to read with/after Tu’s. Its exercises are quite solvable and I learned a lot from it.
In the end, my advise is to get Tu’s and if you feel comfortable after a while with it and want to learn more on the geometry of manifolds, get Nicolaescu’s (or Lee’s).
If you look for an alternative to Tu’s I believe the best one is John M. Lee – “Introduction to Smooth Manifolds”; it is a well-written book with a slow pace covering every elementary construction on manifolds and its table of contents is very similar to Tu’s. Other alternative maybe Boothby – “Introduction to Differentiable Manifolds and Riemannian Geometry” since it also builds everything up starting from multivariable analysis. If you prefer a transition from differential curves and surfaces focusing on riemannian geometry you have Kühnel – “Differential Geometry: Curves, Surfaces, Manifolds“.
Introduction to Smooth Manifolds by John M. Lee is a great text on the subject. It covers similar material to Loring W. Tu’s text. Lee’s book is big (~650 pages) but the exposition is clear and the book is filled with understandable examples.
However, I would argue that one of the best introductions to manifolds is the old soviet book published by MIR, Mishchenko/Fomenko – “A Course of Differential Geometry and Topology”. It develops everything up from RnRn, curves and surfaces to arrive at smooth manifolds and LOTS of examples (Lie groups, classification of surfaces, etc). It is also filled with LOTS of figures and classic drawings of every construction giving a very visual and geometric motivation. It even develops Riemannian geometry, de Rham cohomology and variational calculus on manifolds very easily and their explanations are very down to Earth. If you can get a copy of this title for a cheap price (the link above sends you to Amazon marketplace and there are cheap “like new” copies) I think it is worth it. Nevertheless, since its treatment is a bit dated, it lacks the kind of hard abstract algebraic formulation used nowadays (forget about functors or exact sequences, like Tu or Lee mention), that is why I believe an old fashion geometrical treatment may be very helpful to complement modern titles for a person entering the subject needing a good geometrical foundation. In the end, we must not forget that the old masters that founded the subject were much more visual an intuitive than the modern abstract approaches to geometry, and that motivation was what culminated in the unified abstract approach of nowadays.
Quang Huy 