I apologize for the self promotion -- I hope the content of this answer can be useful anyway...
My favourite introduction to L-theory is Lurie's notes on Algebraic L-theory and surgery (warning: aggressively modern).
Working from the ideas in this notes my coauthors and I have built a series of papers (I, II, III) developing L-theory and hermitian K-theory in a fashion that closely mirrors the development of algebraic K-theory. One of the main point of our work is to find a home for all (well, almost all) variants of L-theory in a single unified framework, that of Poincaré structures.
These papers are probably not very good for a student though -- while we do all the details, this results in many pages of technical results and it's easy to lose the forest for the trees. If you are interested in this circle of ideas some of my coauthors have given a series of minicourses on the topic (Fabian Hebestreit's I, II, Markus Land I, II, III, IV; Yonatan Harpaz I, II, III) which are probably more accessible than the papers.