I am not sure whether it is in the spirit of the original question, but let me add a wordy version of Theo's extensive and excellent bibliography -- a bit more of a road map. Let me divide to this purpose chromatic homotopy theory pseudo-historically in different phases. The years denote the "main phase"; in each case developments have continued after.
Phase 0: Prehistory -- MU and the image of J (1960 - 1965)
For the computation of $\pi_*MU$, Switzer's Algebraic Topology and Lurie's Chromatic homotopy theory are good, for example. For the image of $J$, the classic paper On the groups J(X) IV by Adams a still good to have a look at.
Phase 1: Adams-Novikov spectral sequence, formal groups and greek letters (1967- 1977) This was started by Novikov's introduction of the Adams-Novikov spectral sequence; note that in the same paper a link to formal groups was already established in an appendix by Mischenko! The main points of this phase were to develop a structure theory of BP using formal groups to do computations in the Adams-Novikov spectral sequence, in particular in relation to the elements produced by Smith-Toda complexes ($\alpha, $\beta$- and $\gamma$-family).
- The most comprehensive reference for this is Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres, Douglas C. Ravenel.
- For the structure theory of $MU_*MU$ and $BP_*BP$ it is also good to look at Part II of Adams's Stable Homotopy and Generalized Homology and at Wilsons's Brown-Peterson Homology: Introduction and sampler.
- A shorter overview to the computational aspects is also contained in Ravenel's A novice guide to the Adams-Novikov spectral sequence
- The definite article from this era is Miller-Ravenel-Wilson Periodic phenomena in the Adams-Novikov spectral sequence (1977)
- For Smith-Toda complexes it is also good to look at the much later paper The Smith-Toda complex $V((p+1)/2)$ does not exist by Nave.
- Also have a look at Goerss's The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres
Phase 1b: Unstable computations of homology with respect to $MU$ and $K(n)$ (1973-1980) This was started in Wilson's thesis and two of the main paper's of this part are the Ravenel-Wilson papers The Hopf ring for complex bordism (1977) and The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture (1980). Later expositions include Wilson's BP-sampler mentioned above and Section 2 of Hopkins-Lurie's Ambidexterity paper A nice addition to the topic was the 1994 Hopkins-Hunton paper On the structure of spaces representing a Landweber exact cohomology theory. Hill and Hopkins have also a project, extending some of the results to a $C_2$-equivariant setting.
Phase 2: Large scale phenomena (1979-1988)
This is dominated by the theory of Bousfield localization and the Ravenel conjectures. This is well explained in Lurie's chromatic homotopy theory notes. Also Ravenel's paper Localization with respect to periodic homology theories is still a very good read. Other sources include Ravenel's "orange book" and the original papers by (Devinatz), Hopkins and Smith.
Phase 3: Designer spectra and new computations
The crucial step was here the Goerss-Hopkins-Miller theorem about the action of the Morava stabilizer group on the Lubin-Tate spectra (aka Morava E-theory). This allowed to define higher real K-theory and a little later also $TMF$. The higher real K-theories were used by Goerss-Henn-Mahowald-Rezk (and many others) to write down resolutions for $K(2)$-local spheres and many explicit computations were (and are) done for and using higher real K-theories. For higher real K-theories and the like some good sources are:
- Rezk: Notes on the Hopkins-Miller theorem
- Goerss-Hopkins: Moduli Spaces of Commutative Ring Spectra (last section)
- Goerss, Henn, Mahowald, Rezk: A resolution of the K(2)-local sphere at the prime 3
Phase 3b: Unstable telescopic homotopy theory (1982 - )
There is a notion on $v_n$-periodic unstable homotopy groups. For a survey of the (computational aspects of the) older literature, have a look at Davis's article in the Handbook of Algebraic Topology. Davis, Thompson and especially Mahowald were maybe the founders of the theory, but Bousfield did an enormous amount of work here, culminating in On the $2$-primary $v_1$-periodic homotopy groups of spaces. For the modern aspects, the notes from the Thursday seminar contain a very good treatment. There are also at least two papers of Kuhn that are recommended in this context:
Phase 4: Entering (derived) algebraic geometry (?-?)
In principle, this phase begins with the insights of Morava around 1970. These were translated by Wilson and so into more traditional language. But the insight came back by the introduction of the language of stacks into algebraic topology. Some classic sources here are:
- Hopkins: COCTALOS
- Naumann: The stack of formal groups in stable homotopy theory
This was much enhanced in the construction of $TMF$, where a sheaf of $E_{\infty}$-ring spectra was constructed on the moduli stack of elliptic curves. Theo has already linked some of the best sources for this, but I want to add Goerss's surveys, in particular his Bourbaki presentation. The derived algebraic geometry perspective was of course taken much more seriously in Lurie's approach, with the relevant articles again linked in Theo's answer.
A lot more could (and should) be said, especially about applications to more classical problems. The Kervaire invariant 1 problem uses some serious Phase-1 chromatic homotopy theory. Topological modular forms have been applied to construct new classes in the cokernel of $J$ and thus to obtain new exotic spheres (see e.g. the article of Wang-Xu and the preprint of Behrens-Hill-Hopkins-Mahowald). $tmf$ has also been used to obtain new results on generalized Smith-Toda complexes and thus to a better understanding of the Adams-Novikov 2-line (see the HHA article by Behrens-Hill-Hopkins-Mahowald and the Hopkins-Mahowald article published in the $TMF$-book).