I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks in the subject. Let me mention two of them:
 
- *Stable homotopy and generalized cohomology* by J. Frank Adams is an old classic. Its treatment of some topics is far from modern though, and in particular the development of localizations is flawed and should be complemented by reading Bousfield's original papers. He also doesn't talk about commutative (i.e. $E_∞$) ring spectra, which you're going to need.

- [*Symmetric spectra*](http://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf) by S. Schwede. A much more modern approach, covering a variety of topics you're going to need. Just don't get too hung on the model categorical subtleties of the model he chose (I'm thinking mainly semistability here), 'cause they won't come up in practice.

When you have a sufficiently good background in homotopy theory that the words *spectra* and *group completion* don't make you scream in terror, it's time to start with actual algebraic K-theory. Here are some useful starting points

- [*The K-book*](http://sites.math.rutgers.edu/~weibel/Kbook.html) by Charles Weibel has a lot of classical material and it is a useful bridge from the low dimensional, hand-defined groups to the more modern algebraic K-theory spectrum. It is a bit long though, and I'd treat it more as a  reference than a book to be read from top to bottom.

- [*Higher Algebraic K-theory of Schemes and of Derived Categories*](https://www.gwern.net/docs/math/1990-thomason.pdf) by Thomason and Trobaugh. One of the fundamental papers on algebraic K-theory. It also has a decent introduction to Waldhausen's S-construction and it is worth reading in full. Highly highly recommended.

- [*On the Lichtenbaum-Quillen Conjectures from a Stable Homotopy-Theoretic Viewpoint*](https://link.springer.com/chapter/10.1007/978-1-4613-9526-3_7) by Stephen A. Mitchell. This is a general survey of algebraic K-theory. It is extremely useful and will acquaint you with most classical theorems that you might find used in more specialized papers. Absolutely on a to-read list.

- [*Algebraic K-theory and traces*](http://math.mit.edu/~nrozen/juvitop/madsen.pdf) by I. Madsen. Filling a gap in the previous survey, this talks about trace methods, the best way we have to actually compute the K-theory groups.

- [*Algebraic K-theory of spaces*](https://pub.uni-bielefeld.de/publication/1782197) Waldhausen's original paper on his approach to algebraic K-theory. It's worth taking a look at some of the proofs, and the motivation is explained clearly and naturally.

I also think some familiarity with ∞-category theory might be useful from a technical standpoint, but you shouldn't enter into a full dive into the foundations. This is something best coordinated with your advisor, who will know how much of it is actually useful for you. As a rule of thumb, if you're spending more than a week on it (at first, of course), you're doing too much.

Feel free to jump into the [homotopy theory chatroom](https://chat.stackexchange.com/rooms/9417/homotopy-theory) to hang around and ask questions if you feel like it.