I'm looking for some references for an introduction to the Luttinger liquid.
For a shorter introduction, providing intuition and highlighting the differences between Fermi and Luttinger liquids, I'd recommend T. H. Hansson's Fermi and Luttinger liquids lecture notes.
I've linked the version for the 2018 Nordita winter school, which appears to be an expanded version of the 2011 version I've found useful myself. Note that this isn't a rigorous or comprehensive introduction to one-dimensional fermion systems. Instead, Hansson moves rather quickly from bosonization to fractional quantum Hall systems. His treatment is clearly inspired by Schultz et al.'s review paper, to which I'll return below.
A good textbook introduction and overview is found in Chapter 6 of Eduardo Fradkin's Field Theories of Condensed Matter Physics (Cambridge University Press). This is perhaps a more traditional presentation than Hansson's, starting from quantum wires, and introducing charge & spin density waves in more detail. Fradkin also discusses susceptibilities, e.g. in the context of ARPES and STM experiments.
If you want to dig into the details, the standard reference would be Thierry Giamarchi's Quantum Physics In One Dimension (Oxford University Press). This book covers bosonization, Luttinger liquids, spin systems, analytical methods, and more. This is more of a monograph than an easily digested textbook, the "Mahan of Luttinger liquids", if you will. However, Giamarchi's writing is clear, and particularly the first few chapters are quite accessible.
As mentioned above, there is a set of lecture notes from Schultz, Cuniberti and Pieri: Fermi liquids and Luttinger liquids from the 1997 Chia Laguna summer school. Compared to Hansson's notes, there is more focus on transport and spin systems (chains and ladders).
Philip Phillips' book Advanced Solid State Physics (Cambridge) contains a shorter discussion of bosonization and Luttinger liquids. In my opinion, this is more of an appetizer than a comprehensive introduction.
Gogolin, Nersesyan and Tsvelik has a book called Bosonization and Strongly Correlated Systems (Cambridge). It does contain a fairly technical field-theoretical treatment of Luttinger liquids, but I'd tend to recommend Giamarchi's book first unless you prefer more of a high-energy perspective.
If you want to venture beyond the scope of Luttinger liquid theory, consider Imambekov, Schmidt, and Glazman's review paper One-Dimensional Quantum Liquids: Beyond the Luttinger Liquid Paradigm. It might not be where you want to start, but it's interesting to know about it.
The bosonization literature in particular suffers from a bad case of people using different notations. While I've provided several resources, chances are that you'd want to work through one at a time in order to avoid unnecessary confusion. Appendix D of Giamarchi's book actually provides a notation dictionary to ameliorate this problem.