There are two(or three maybe) way to go to the topological K-theory, one is from the algebraic topology(or vector bundles), the other is from(download) the operator K-theory(the K-theory of C*-algebras).
Form the algebraic topology: there are many second course book mention it, for example:
May J P. A concise course in algebraic topology[M]. University of Chicago Press, 1999.
Switzer R M. Algebraic topology--homotopy and homology[M]. Springer, 1975.
Aguilar M, Gitler S, Prieto C. Algebraic topology from a homotopical viewpoint[M]. Springer Science & Business Media, 2008.
From the vector bunddle:
Hatcher A. Vector bundles and K-theory[J]. Im Internet unter http://www.math.cornell.edu/~hatcher, 2003.
D. Husemoller, Fibre bundles. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994.
There is also an online course note:Algebraic Topology II: Topological K-Theory (Spring 2015)
From the operator K-theory(K-theory of C*-algebras): maybe the only one is:
Park E. Complex topological K-theory[M]. Cambridge University Press, 2008.
Some K-theory of C*-algebras books also mention a little topological K-theory as a background, you can see this book:
Blackadar B. K-theory for operator algebras[M]. Cambridge University Press, 1998.
I am making some videos of K-theory(from topological to operator) in my language Chinese, if you can read Chinese or have some friend help to translate, you can see them in my blog: