Ib Madsen's survey
MR1474979 (98g:19004) Madsen, Ib Algebraic K-theory and traces.
Current developments in mathematics, 1995 (Cambridge, MA), 191–321,
Int. Press, Cambridge, MA, 1994.
was written to be such an introduction, but is 25 years old. Its technical foundations are given in terms of FSPs (functors with smash product). A more recent source is the book
MR3013261 Dundas, Bjørn Ian; Goodwillie, Thomas G; McCarthy, Randy
The local structure of algebraic K-theory. Algebra and Applications,
18. Springer-Verlag London, Ltd., London, 2013. xvi+435 pp. ISBN: 978-1-4471-4392-5; 978-1-4471-4393-2
which works with $\Gamma$-rings (= monoids in $\Gamma$-spaces). Both of these predate the connection to crystalline and syntomic cohomology.