To dash off a quick answer, Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in Grothendieck's letters to Larry Breen from 1975, and is mostly contained in the letter to Quillen which makes up the first part of PS (about 12 pages or so). Maltsiniotis has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.
The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (see there for extensive literature relating to derivators, the first 15 of 19 chapters of Les Dérivateurs are themselves available), and the 'schematisation of homotopy types', which is covered by work of Toën, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk and others dealing with derived algebraic geometry, which is going far ahead of what I believe Grothendieck envisaged.
During correspondence with Grothendieck in the 80s, Joyal constructed what we now call the Joyal model structure on the category of simplicial sets simplicial sheaves to give a basis to some of the ideas being tossed around at the time. (Edited 2022)
Edit: I forgot something that is in PS, and that is the theory of localisers and modelisers, Grothendieck's conception of homotopy theory which you mention, which is covered in the work of Cisinski.
Edit 2019: Toën has a new preprint out
Bertrand Toën, Le problème de la schématisation de Grothendieck revisité, arXiv:1911.05509
with abstract starting
"The objective of this work is to reconsider the schematization problem of [Pursuing Stacks], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6 of Toën's Champs affines]..."
I would think this is from an authoritative source, since apparently the author consulted with Bourguignon (chair of the hiring committee at CNRS).
When Grothendieck reapplied to the CNRS in 1984, his application was
once again controversial. Jean-Pierre Bourguignon, now director of the
IHÉS, chaired the committee in charge of reviewing applications in
mathematics, among which was Grothendieck’s. According to Bourguignon,
in the handwritten letter required for the application, Grothendieck
listed several tasks he would not perform, such as supervising
research students. Because CNRS contracts obligate researchers to
perform some of these tasks, this letter was viewed by the CNRS
administration as proof of Grothendieck’s ineligibility. Bourguignon
said he tried to get Grothendieck to amend his application so that it
did not state explicitly all the tasks he refused to carry out, but
Grothendieck would not budge. After considerable effort on the part of
several people, Grothendieck was eventually put on a special kind of
position, called a position asterisquée, that was acceptable to him
and to the CNRS. The CNRS did not actually hire him but was in charge
only of paying his salary, and he retained his university
affiliation. So for his last few years at Montpellier before his
retirement in 1988, Grothendieck did not teach and spent less and less
time at the university.
Best Answer
Really Martin's comment should be the answer, but note Wikipedia also gives:
Alexander Grothendieck, 1984. "Esquisse d'un Programme", (1984 manuscript), finally published in Schneps and Lochak (1997, I), pp.5-48; English transl., ibid., pp. 243-283.
and the latter reference is :
Schneps, Leila; Lochak, Pierre, eds. (1997), Geometric Galois Actions I: Around Grothendieck's Esquisse D'un Programme, London Mathematical Society Lecture Note Series, 242, Cambridge University Press, ISBN 978-0-521-59642-8
(This even has an MR number!)