There are no books that you must read but here are selection of books that you might find helpful.
Topology (2nd edition) by J.R. Munkres (516.11 MUN
SHORT-LOAN).
I have not read this but it comes highly recommended by previous
lecturers of this unit. It is supposed to cover all the course
material and more.
Topology : a first course by J.R. Munkres (516.11
MUN).
First edition of the above: still good but does not have the
classification of surfaces.
Introduction to Metric and Topological Spaces by
W.A. Sutherland (513.22 SUT).
One of my favourites! Dead clear but does not do the
classification of surfaces.
Basic topology by M.A. Armstrong (516.11
ARM).
Another of my favourites.
Algebraic topology: an introduction by W.S. Massey
(516.12 MAS).
A bit advanced but the first chapter gives an account of the
classification of surfaces.
Counterexamples in topology by L.A. Steen and J.A. Seebach
(516.113 STE ).
If you need a counter-example or just want to contemplate the weird
spaces that exist, this is the book for you.