Math 3760: Topics in Topology
Here is the firstday
handout.
General References
 Differential Topology, V. Gullemin and A. Pollack. "Friendly"
and a nice read.
 Foundations of Differentiable Manifolds and Lie Groups, F.
Warner. Less friendly. Terse to a fault and lacking
examples in places, it also does everything "right" (a doubleedged sword,
but very useful for referencing).

Algebraic Topology, Allen Hatcher. The goto algebraic topology
reference for the nonalgebraic topologist.
It is freely available on his website (and linked above).
 The standard reference for virtually anything in PL topology is
Rourke & Sanderson's Introduction to
PiecewiseLinear Topology.
More accessible (also less general and less formal) is M.A. Armstrong's
Basic
Topology, which also has lots of other good stuff.
Project Possibilities
These papers/book sections are "reasonably" approachable
(to varying degrees).
 For the classification of topological 1manifolds:
The Classification
of 1Manifolds: a TakeHome Exam. David Gale, The American
Mathematical Monthly.
 For the classification of smooth (compact) 1manifolds: Appendix 2
of Guillemin & Pollack.
 For existence and uniqueness of smooth structures on surfaces:
The Kirby torus trick for
surfaces. Allen Hatcher, arXiv:1312.3518.
 The first "exotic'' smooth structures, on the 7sphere:
On Manifolds Homeomorphic to
the 7sphere. J. Milnor, Annals of Mathematics, 1956.
 There is a nice explanation of the Jordan separation theorem, the Jordan
curve theorem, and the Schoenflies
theorem in Munkres' Topology.
For an algebraic topology proof of the higherdimensional Jordan
separation
theorem, and an elementary
construction of the Alexander horned sphere, see Appendix 2.B of
Hatcher's
Algebraic Topology.
 A proof of the
generalized Schoenflies theorem, M. Brown. Bulletin of the AMS,
1960.
The title says it all. I have not read this proof but it is amazingly
short (3 pp) and selfcontained (1 reference).
Further Reading
 Boy's surface
(at Wikipedia): an immersion of RP^2, the real projective plane, into R^3.

Lectures on Diffeomorphism Groups of Manifolds. Alexander Kupers.
There's a whole lot here, with heavy machinery, but the beginning is nicely
written and approachable.
 Groups of Homotopy Spheres: I.
M. Kervaire and J. Milnor, Annals of Mathematics, 1962.
Describes a group structure (with operation connected sum) on the set of
exotic spheres up to
diffeomorphism in dimensions at least 5 (after the
hcobordism
theorem).
 The math review
(subscription needed) of J. Cerf's proof (in a series of
four substantial papers written
in French) that "\Gamma_4=0", ie. that
any two OP diffeomorphisms of S^3 are smoothly isotopic.
 Stable homeomorphisms and
the annulus conjecture. R. Kirby, Annals of Mathematics, 1969.
A proof of the Stable
Homeomorphism Conjecture, hence also the Annulus Conjecture, in dimensions
at least 5. This implies by standard arguments
that any two OP homeomorphisms of S^n are isotopic.

Extending Diffeomorphisms, R. Palais. Proceedings of the AMS, 1960.
Theorem B here implies that the connected sum of smooth manifolds
is well defined. The corresponding
result for topological manifolds
uses the Annulus Conjecture.
 The
Hauptvermutung Book, ed. A.A.Ranicki.
Actually a collection of papers disproving the "hauptvermutung der
kombinatorischen topologie", on
homeomorphic simplicial complexes being
isomorphic after subdivision. Ranicki collected them and
wrote a nice
introduction stating the problem and giving context.