That depends on your mathematical background and the level of abstraction you want.
The minimum amount of concepts you should be familiar with, ranges from topological spaces and its basic properties (elements of point-set topology) to multivariable calculus (covering implicit and inverse function theorems), passing by elements of multilinear algebra to manage tensors. A good mathematical physics book where you can find this and even more, like an introduction to differential geometry through manifolds, is Szekeres' "A Course in Modern Mathematical Physics".
If you need this little background in topology, very good and short books on general topology are Runde's "A Taste of Topology" for a more formal approach and Jänich's "Topology" for a more didactic and graphical-reasoned introduction. Then the approach to manifolds from pure differential topology could be started by Jänich - "Introduction to Differential Topology" which is very graphical, short enough and quick to the point to master fully.
But if you are really interested in these matters from a theoretical physics perspective, you should by all means read the book by Nakahara - "Geometry, Topology and Physics"; there you can read a very thorough introduction (although sometimes concise) to most of the topics of differential geometry and topology of interests for physics: homology and homotopy groups, calculus on manifolds, Riemannian geometry, complex geometry, fiber bundles, connections, characteristic classes, index theorems with applications. It is a wonderful book if you have enough background and use supplementary readings (for example in tensor, multilinear, algebra). Other similar book with the same spirit is Frankel's "The Geometry of Physics" which is longer dealing with most of the same contents more deeply; nevertheless I do not find it as useful and straight to the point as Nakahara's (and Frankel's notation seems to me not the most orthodox standard compared to other books I have used on the subject).
Similarly, I recommend the new book by Eschrig - "Topology and Geometry for Physics" which again deals with the same content as Nakahara's but with a less mathematical exposition (definition, theorem, proof...) since it is written to be read as lectures or a physics text. However, it is very detailed and instructive anyway, uses many figures and develops the same amount of detail or more at some points. This kind of books develop the necessary topological background all along as needed so if you have enough background on vector analysis/multivariable calculus you can approach these books directly to learn about manifolds.
For a more mathematical purely formal treatment of differential geometry on manifolds I would dive in the wonderful book by Nicolaescu - "Lectures on the Geometry of Manifolds" since it is very complete and modern. Another slower mathematical exposition is Jeffrey Lee's "Manifolds and Differential Geometry" which may be useful to you as a companion to the other physics-oriented texts, since it develops many details and background at some points. They are both wonderful books in my opinion.
For an approach to manifolds through mechanics, the classic book of V.I. Arnol'd - "Mathematical Methods of Classical Mechanics" is great but it is very applied and focused on symplectic geometry (and despite being a masterpiece, it is a little bit out-dated in style for my tastes). Another differential-geometric introduction to mechanics is José/Saletan's "Classical Dynamics: A contemporary approach". The most advanced book of this kind is the bible by Marsden/Abraham - "Foundations of Mechanics" which I think is a masterpiece trying to develop the complete theory of classical mechanics (hamiltonian/lagrangian/Hamilton-Jacobi) from a purely differential-geometric perspective using manifolds.
My personal advise would be to read Eschrig alongside Nakahara (or Frankel depending on your likings of style), that is what I did with a background on theoretical physics. Then you should consult some of the other books to fill particular gaps and needed background or to use as companions for more examples or exercises.
This is a wonderful and very interesting subject, good luck!