Refrences for Differentiable Manifolds:</br>
</br>
The easiest introductions are:
<ul>
 <li> Lee
  <i> Introduction to Smooth Manifolds</i>
   Springer 2003
 <li> Boothby
  <i> An introduction to differentiable manifolds and Riemannian geometry</i>
  Acad. Press 2002
 <li> Spivak
  <i> A Comprehensive Introduction to Differential Geometry Vol I-II</i>
  Publish & Perish 1999
</ul>
Some further classical differentiable manifold references are:
<ul>
 <li> Lang
  <i> Introduction to Differential Manifolds</i> Springer Verlag 2002
 <li> Warner
  <i> Foundations of Differentiable Manifolds and Lie Groups.</i>
      Springer Verlag 1983
 <li> Dubrovin, Formenko, Novikov 
      <i>Modern geometry-methods and applications vol I-II</i>
      Springer 1992 
</ul>
Spivak and Lee are extensive whereas Lang and Warner are concise.
Dubrovin et al starts with local th.
<br>
<br>
A couple of classical undergraduate texts are
<ul>
<li> Spivak <i> Calculus on Manifolds</i> Benjamin 1965
<li> Munkers <i> Analysis on Manifolds</i> Westview Press 1991
</ul>
A couple of references that also cover applications to physics:
<ul>
 <li> Abraham, Marsden, Ratiu
   <i> Manifolds, Tensor Analysis, and Applications</i>
   Springer Verlag 1988
 <li> Darling
  <i> Differential Forms and Connections</i>
  Cambridge University Press 1994
  <li> Schutz <i> Geometgrical methods of mathematical physics</i>
   Cambridge University Press 1980
  <li> Frankel <i> The Geometry of Physics: An itroduction</i>
   Cambridge University Press 2003 
</ul>
Abraham et al is an extensive source on manifold theory including
mentioned applications.
Darling does things more concretely,
with applications to electromagnetism,
but it doesn't cover the core syllabus.