ADDITION: I have compiled what I think is a definitive collection of listmanias at Amazon for a best selection of books an references, mostly in increasing order of difficulty, in almost any branch of geometry and topology. In particular the books I recommend below for differential topology and differential geometry; I hope to fill in commentaries for each title as I have the time in the future.
If you want to have an overall knowledge Physics-flavored the best books are Nakahara's "Geometry, Topology and Physics" and above all: Frankel's "The Geometry of Physics" (great book, but sometimes his notation can bug you a lot compared to standards).
If you want to learn Differential Topology study these in this order: Milnor's "Topology from a Differentiable Viewpoint", Jänich/Bröcker's "Introduction to Differential Topology" and Madsen's "From Calculus to Cohomology". Although it is always nice to have a working knowledge of general point set topology which you can quickly learn from Jänich's "Topology" and more rigorously with Runde's "A Taste of Topology".
To start Algebraic Topology these two are of great help: Croom's "Basic Concepts of Algebraic Topology" and Sato/Hudson "Algebraic Topology an intuitive approach". Graduate level standard references are Hatcher's "Algebraic Topology" and Bredon's "Topology and Geometry", tom Dieck's "Algebraic Topology" along with Bott/Tu "Differential Forms in Algebraic Topology."
To really understand the classic and intuitive motivations for modern differential geometry you should master curves and surfaces from books like Toponogov's "Differential Geometry of Curves and Surfaces" and make the transition with Kühnel's "Differential Geometry - Curves, Surfaces, Manifolds". Other nice classic texts are Kreyszig "Differential Geometry" and Struik's "Lectures on Classical Differential Geometry".
For modern differential geometry I cannot stress enough to study carefully the books of Jeffrey M. Lee "Manifolds and Differential Geometry" and Liviu Nicolaescu's "Geometry of Manifolds". Both are deep, readable, thorough and cover a lot of topics with a very modern style and notation. In particular, Nicolaescu's is my favorite. For Riemannian Geometry I would recommend Jost's "Riemannian Geometry and Geometric Analysis" and Petersen's "Riemannian Geometry". A nice introduction for Symplectic Geometry is Cannas da Silva "Lectures on Symplectic Geometry" or Berndt's "An Introduction to Symplectic Geometry". If you need some Lie groups and algebras the book by Kirilov "An Introduction to Lie Groops and Lie Algebras" is nice; for applications to geometry the best is Helgason's "Differential Geometry - Lie Groups and Symmetric Spaces".
FOR TONS OF SOLVED PROBLEMS ON DIFFERENTIAL GEOMETRY the best book by far is the recent volume by Gadea/Muñoz - "Analysis and Algebra on Differentiable Manifolds: a workbook for students and teachers". From manifolds to riemannian geometry and bundles, along with amazing summary appendices for theory review and tables of useful formulas.
EDIT (ADDED): 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 $\mathbb{R}^n$, 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, the kind of algebraic formulation is not used (forget about pullbacks and functors, like Tu or Lee mention), that is why an old fashion geometrical treatment may be very helpful to complement modern titles. In the end, we must not forget that the old masters were much more visual an intuitive than the modern abstract approaches to geometry.
NEW!: the book by Mishchenko/Fomenko, along with its companion of problems and solutions, has been recently typeset and reprinted by Cambridge Scientific Publishers!
If you are interested in learning Algebraic Geometry I recommend the books of my Amazon list. They are in recommended order to learn from the beginning by yourself. In particular, from that list, a quick path to understand basic Algebraic Geometry would be to read Bertrametti et al. "Lectures on Curves, Surfaces and Projective Varieties", Shafarevich's "Basic Algebraic Geometry" vol. 1, 2 and Perrin's "Algebraic Geometry an Introduction". But then you are entering the world of abstract algebra.
If you are interested in Complex Geometry (Kähler, Hodge...) I recommend Moroianu's "Lectures on Kähler Geometry", Ballmann's "Lectures on Kähler Manifolds" and Huybrechts' "Complex Geometry". To connect this with Analysis of Several Complex Variables I recommend trying Fritzsche/Grauert "From Holomorphic Functions to Complex Manifolds" and also Wells' "Differential Analysis on Complex Manifolds". Afterwards, to connect this with algebraic geometry, try, in this order, Miranda's "Algebraic Curves and Riemann Surfaces", Mumford's "Algebraic Geometry - Complex Projective Varieties", Voisin's "Hodge Theory and Complex Algebraic Geometry" vol. 1 and 2, and Griffiths/Harris "Principles of Algebraic Geometry".
You can see their table of contents at Amazon.
Hope this helps... good luck!
You shouldn't need much. Almost all you need to know about topology (especially of the point-set variety) should have been covered in a course in advanced calculus. That is to say, you really need to know about "stuff" in $\mathbb{R}^n$. (The one main exception is when you study instantons and some existence results are topological in nature; for that you will need to know a little bit about fundamental groups and homotopy.) The reason is that differential topology and differential geometry study objects which locally look like Euclidean spaces. This dramatically rules out lots of the more esoteric examples that point-set topologists and functional analysts like to consider. So most introductory books in differential geometry will quickly sketch some of the basic topological facts you will need to get going.
In terms of topology needed for differential geometry, one of the texts I highly recommend would be
It is quite mathematical and quite advanced, and covers large chunks of what you will call differential geometry also. One can complement that with his Riemannian Manifolds to get some Riemannian geometry also.
But since you are asking from the point of view of a Physics Undergrad, perhaps better for you would be to start with either (or both of)
and follow-up with
Best Answer
By ``basic topology of $\mathbb{R}^n$'' I assume that you are familiar with the notions of openness, closedness, connectedness, and compactness. If you are unclear on these notions (I found compactness hard to get used to), you should remedy that before attempting to learn differential geometry.
If you understand these, then you're probably already prepared to read an introductory book on differential geometry, such as do Carmo's Differential Geometry of Curves and Surfaces or O'Neill's Elementary Differential Geometry. Apart from the concepts I mentioned above, all the necessary topology is developed alongside the geometry in these books (e.g. homeomorphism, homotopy, Euler characteristic, and so on).
If you want to learn quickly about the topology of smooth manifolds without having to learn about general topological spaces, there is probably no better place to look than Milnor's Topology from the Differentiable Viewpoint. A more in-depth treatment along the same lines is Guillemin and Pollack's Differential Topology.