K-theory
M. F Atiyah, D.W. Anderson, 1967.

K-theory and the Hopf invariant
J. F. Adams, M. F. Atiyah, 1966.
This gives a short proof to Adams’s long Annals paper result on the Hopf invariant 1 using the Adams spectral sequence.

Vector fields on spheres (TeX notes), original is Part 1 and Part 2
Course taught by Haynes Miller, 1989.

Topological K-theory and some of its applications
Saad Slaoui, 2019.

Blog post on K-theory and the Hopf invariant
Akhil Mathew, 2012.

18 Lectures on K-theory
Ioannis P. Zois, 2010.
Lecture notes from an Introduction to K-theory course covering topological K-theory, analytic K-theory and higher algebraic K-theory.

The Index of Elliptic Operators, I
Atiyah and Singer, 1968.
This paper gives a proof of the Atiyah-Singer index using K-theory.

The Index of Elliptic Operators: II
Michael Atiyah, Graeme Segal, 1968.
This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K-theory. This makes it more challenging on first approach.

K-theory and elliptic operators
Gregory D. Landweber, 2005.
An expository account.