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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 …

You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least …

I am not sure if this is going to be a real answer to the question. However I believe these observations might be interesting. Let me briefly sketch a way to describe a $G$-structure in (excessively) …

The easy way for me to think about this things is via the Yoneda lemma. This works better with reduced $K$-theory. This is defined for a pointed space $X$ as $$ \tilde K^0 (X) := ker(K^0(X)\to K^0(*)) …

Ravenel in his Complex Cobordism and Stable Homotopy Groups of Spheres attributes this result to Stong, in Determination of $H^*(BO(k,⋯,∞),Z_2)$ and $H^∗(BU(k,⋯,∞),Z_2)$, but looking at that paper (wh …

I claim that for every $A_\infty$-space $A$, there is a canonical $A_\infty$-ring structure on $\Omega^\infty\Sigma^\infty_+A$. First, $\Sigma^\infty_+$ from spaces to spectra is symmetric monoidal. …

As a first introduction I like these notes by Achim Krause and Thomas Nikolaus. They do require some familiarity with spectra and stable homotopy theory though.

Since the question remains unanswered, let me copy Tom Goodwillie's comment: If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely g …

I do not know of an answer for a general triangulated category (non-topological triangulated categories are very unusual), but as soon as you ask for some more structure the thesis follows very quickl …