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.