Such a timeline is necessarily highly subjective.
With this disclaimer in mind, we can identify some important turns in the development of foundations of homotopy theory.
The list below concentrates on developments that in some way affect the foundations of homotopy theory, as opposed to general advances in homotopy theory.
Given the length of the list, I probably omitted many important developments,
feel free to point them out in the comments!
I also excluded from consideration the last decade or so, restricting to older developments.
Poincaré defined homology (via Betti numbers) and the fundamental group in a series of papers starting from 1895.
The initial approach was nonrigorous, but in response to the resulting criticism,
Poincaré reformulated his work in terms of simplicial complexes.
Fréchet defined metric spaces in 1906 and Hausdorff defined topological spaces in 1914.
This enabled to study the topological properties of spaces without first triangulating them.
Around 1925, Emmy Noether proposed to upgrade Betti numbers to homology groups.
In connection with this, sometime in 1930s, the terminology shifted from “combinatorial topology” to “algebraic topology”.
Around 1931, Veblen and J. H. C. Whitehead introduced the modern definition of a smooth manifold.
Eilenberg defined singular homology in 1943,
which resulted in a systematic study of homology and cohomology (defined by Kolmogoroff and Alexander in 1936)
of arbitrary topological spaces.
Around 1945, Leray introduced sheaves and spectral sequences. The relevant theory was further developed by Cartan, Serre, and others.
Eilenberg and MacLane introduced categories, functors, and natural transformations in 1945.
Ever since then, category theory played an increasingly important role in homotopy theory,
to the point where we are now often unable to cleanly separate them.
Eilenberg and Zilber developed the theory of simplicial sets (known at the time as “complete semi-simplicial complexes**) in 1949.
J. H. C. Whitehead proved what is now known as the Whitehead theorem in 1948.
Eilenberg and Steenrod published their Foundations of Algebraic Topology in 1952,
formulating what is now known as the Eilenberg–Steenrod axioms.
Around 1953, Cartan and Eilenberg completed their book on homological algebra (published in 1956).
Kan (advised by Eilenberg) systematically developed simplicial homotopy theory (and briefly also cubical homotopy theory)
starting from around 1955.
He introduced combinatorial homotopy groups, the Dold–Kan correspondence,
adjoint functors, limits and colimits, Kan extensions, etc.
Lima defined spectra in 1958.
Quillen published his Homotopical Algebra in 1967, introducing model categories
and using them in his Rational homotopy theory around 1968.
Around 1972, he introduced higher algebraic K-theory.
In 1971, Gabriel and Ulmer published their systematic account of locally presentable categories.
Segal introduced Γ-spaces around 1972.
At the same time, May introduced operads, also in connection with infinite loop spaces.
Brown studied the homotopy theory of sheaves of spaces and spectra in 1972.
Boardman and Vogt introduced quasicategories in 1973.
In 1977, Sullivan published his work on rational homotopy theory in the language of commutative differential graded algebras, complementing the previous work by Quillen.
Dwyer and Kan introduced and developed the theory of simplicial localizations starting from around 1979.
Around 1979, Bousfield introduced what is now known as Bousfield localizations.
In 1983, Grothendieck introduced what is now known as Grothendieck homotopy theory, as well as derivators.
In 1980s, Joyal established what is now known as the Joyal model structure on simplicial sets.
In mid-1980s, Segal (following Witten) introduced what is now known as functorial field theory,
later studied by Atiyah, Kontsevich, Freed, Lawrence, and many others.
In 1985, Jardine gave an account of simplicial presheaves.
Around 1986, Lewis, May, Steinberger, McClure introduced genuine equivariant spectra.
In 1989, Makkai and Paré published a systematic account of accessible categories.
In 1995, Baez and Dolan formulated the cobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture
about (∞,n)-categories for arbitrary n.
In 1997, Elmendorf, Kriz, Mandell, May published the first ever account of a symmetric monoidal category of spectra.
In 1998, Hovey, Shipley, Smith published an account of symmetric spectra.
In 1998, Rezk introduced complete Segal spaces.
In the late 1990s, Voevodsky introduced and developed motivic homotopy theory (including some joint work with Morel).
Around the late 1990s, Smith introduced combinatorial model categories
and proved what is now known as the Smith recognition theorem
and established the existence of left Bousfield localizations of left proper combinatorial model categories.
Monoidal model categories were systematically studied by Schwede and Shipley starting from 1997.
In 2006 (based on a 2003 preprint), Lurie's Higher Topos Theory came out, first as an online draft, which was later published.