A vexing part of algebraic topology is the lack of advanced textbooks - we have to make do with papers. This is unfortunate, since it makes the parts of homotopy theory that I find beautiful seem scattered and abstruse. The following is a list of my favorite papers and books. It is far from complete - in particular it is strongly biased towards my own tastes. I hope by maintaining this that people will be inspired to take a look at these classic papers and books.

After a first course

Assuming you've completed algebraic topology from something like Hatcher's Algebraic Topology, Bredon's Geometry and Topology or May's A Concise Course

Mosher and Tangora

Milnor and Stasheff

K-theory

Spectra

A good place to start is with Adams' Lectures on Homology, which is a technical master grappling with spectra.

EKMM Spectra

Diagram Spectra
Mandell, M A, J P May, S Schwede, and B Shipley. 2001. “Model Categories of Diagram Spectra.” Proceedings of the London Mathematical Society 82 (02): 441–512.

Model Categories

A good introduction is
Dwyer and Spalinski
A nice more advanced source is Hirschorn's text. Ponto and May has some excellent material on model categories.
Riehl's book on Categorical Homotopy Theory is also fantastic Finally, appendix of A of Higher Topos theory by Lurie is quite readable.

Abstract Homotopy Theory

Vogt - Homotopy Colimits
Vogt, R M. 1973. “Homotopy Limits and Colimits.” Mathematische Zeitschrift 134 (1): 11–52. This paper basically proves everything you want about homotopy colimits, and it does it cleanly. Even provides a proof of one of Thomason's results

Dwyer-Kan Simplicial Localization
Dwyer, W G, and D M Kan. 1980. “Simplicial Localizations of Categories.” J. Pure Appl. Algebra 17 (3): 267–284.
Dwyer, W G, and D M Kan. 1980. “Calculating Simplicial Localizations.” J. Pure Appl. Algebra 18 (1): 17–35.
Dwyer, W G, and D M Kan. 1980. “Function Complexes in Homotopical Algebra.” Topology 19 (4): 427–440.

K-theory Foundations

The first things one should probably read are
Quillen, D. 1973. “Higher Algebraic K-Theory: I.” Higher K-Theories: 85–147.
and
Waldhausen, F. 1985. “Algebraic K-Theory of Spaces.” Algebraic & Geometric Topology LMN 1126: 318–419
Both of these are papers that I have never stopped learning from - they are an endless source of both insight and tricks.

The Cyclotomic Trace

Category Theory