I will mainly follow the book by Alexandru Scorpan with a possible addition of the work by Fintushel and Stern. I will start from h-cobordism theorem by Smale and the work of Freedman.
After that I will classify quadratic forms and discuss the results of Rokhlin and give explicit examples of non-smoothable manifolds. At the end I will present smooth 4-manifolds   having different smooth structures.

Lecture 1. The Poincare conjecture for dimensions \ge 5.
Lecture 2. Topological 4-manifolds and h-cobordisms.
Casson's handles. Freedman's theorem.
Lecture 3. The intersection form. Signatures. The K3 surface.
Lecture 4. The Whitehead theorem (intersection form defines the
homotopy type of simply connected 4-manifolds).
Lecture 5. Rokhlin's theorem.
Lecture 6. The canonical class. Serre classification of forms.
Lecture 7. Topological classifiaction due to Freedman. Non-realisability of forms by
smooth manifolds. Exotic R^{4}.
Lecture 8. Donaldson's invariants.
Lecture 9. Seiberg-Witten invariants.

Basics of algebra, basics of topology, possibly, some knot theory

[1] A.Scorpan, The wild world of 4-manifolds;
[2] V.O. Manturov, I.M. Nikonov, “On Braids and Groups G_n^k”, Journal of Knot Theory and Its Ramifications, 24:10 (2015), 16 pp.
[3] https://www.researchgate.net/publication/333035589_On_Groups_G_nK_and_G_nk_A_Study_of_Manifolds_Dynamics_and_Invariants