I want to use the technicalities of structured ring spectra for the first time in my life, and I am not really familiar with the relevant literature. I am looking for a reference that defines algebraic K-theory for associative unital algebras in symmetric spectra.



Algebraic $K$-theory of algebras in symmetric spectra: reference
Tom, not precisely sure what you want. I'm guessing you want
to think of a commutative symmetric ring spectrum $R$ and then an
$R$-algebra $A$. Without the extra layer, just using an $S$-algebra $R$
(not necessarily commutative), the original source for the algebraic 
$K$-theory of $R$ is Chapter VI of EKMM, but of course in terms of the
$S$-algebras there. 


The essential point is to have a homotopically
well-behaved symmetric monoidal category of spectra in which to work.
We now have several: symmetric spectra, orthogonal spectra, and EKMM
$S$-modules. Since we have Quillen equivalences that preserve smash
products among them, for many foundational purposes it doesn't matter 
in which category you work. Since that was clear early on, there are
many things in EKMM that have not been repeated in full detail in later
sources. The differences in detail are not especially significant
and comparison theorems are not hard to come by showing that it doesn't
matter where you work. (That is emphatically not true when doing 
multiplicative infinite loop space theory, which is relevant but I 
think not what you are asking.) Probably there is a good more 
recent source focusing on symmetric spectra, but I don't know of one.


Here is a relevant quote from a nice paper of Blumberg and Mandell
(The localization sequence for the algebraic $K$-theory of topological
$K$-theory): "We work in the context of EKMM $S$-modules, $S$-algebras, 
and $R$-modules. Since other contexts for the foundations of a modern 
category of spectra lead to equivalent $K$-theory spectra, presumably 
the arguments presented here could be adjusted to these contexts, but the 
EKMM categories have certain technical advantages that we exploit and that 
affect the precise form of the statements below."


