Everything from Quantum mechanics can be derived from six (?) postulates. Similarly classical electrodynamics can be reduced to Maxwell's equations and Lorentz force law, and special relativity is based on two postulates.
Are there similar sets of postulates for Newtonian mechanics/gravity or other fields of physics, like quantum field theory or string theory?
Many fields of physics are axiomatized, either completely or to some extent.
First of all, for something that will be important for almost all fields, both Lagrangian mechanics and Hamiltonian mechanics are rooted in formal math, via calculus on the jet bundle for the Lagragian, and the Legendre bundle for the Hamiltonian (or for something less complex, Gâteaux derivatives on functionals for the Lagrangian and the Legendre transform for the Hamiltonian). You can check this for instance, as well as Henneaux for all constraint-related matters on the topic.
Special relativity has quite a variety of axiom system, either based on the fairly straightforward theory of Lorentz affine spaces, such as described in Gourgoulhon, or through awful first-order axiom systems such as $\text{Basax}$, $\text{Reich}$ or variations. You can learn more about such axiom systems here for instance. It's also possible to axiomatize it via its causal structure, as done by Zeeman, Carter, Penrose and Kronheimer.
General relativity is also based on axiomatic rules. Basically a spacetime is a tuple $(\mathcal M, \mathcal A, g, \nabla)$, with $\mathcal M$ an $n$-dimensional ($n \geq 2$), Hausdorff, paracompact manifold, $\mathcal A$ a smooth structure on that manifold, $g$ a section of the metric bundle of signature $(-++...)$, and $\nabla$ the Levi-Civitta connection. It's also often assumed to be $(\mathcal M, \mathcal A, g, \nabla, \uparrow, \varepsilon)$, with $\uparrow$ a time-orientation and $\varepsilon$ a measure form. Then you can define the matter content and dynamics on it via sections of vector bundles and the Lagrangian formalism.
Quantum mechanics is usually defined via the Dirac-von Neumann axioms, as a theory of operators acting on a Hilbert space (a good review is in Hall, with a nice overview of the probabilistic shenanigans reviewed in Streater), or via path integrals using the Wick-rotated Wiener functional integral on the configuration space of the system. It is also possible to axiomatize it on less popular mathematical (equivalent) formalisms such as fractional quantum mechanics (where particles are described by stochastic processes), or deformation quantization.
Quantum field theory is harder to axiomatize, but there are a variety of attempts, more or less successful.
All of these are described to some degree in Glimm and Jaffe, as well as Wightman and Streater and Haag. There's a handful of other axiomatizations, such as functorial quantum field theory.
Most of those work fine only really for the free case. There are some attempts at extending those systems to the interacting case as well involving a lot of really awful microlocal analysis and Wick polynomials.
Classical mechanics isn't terribly hard to axiomatize. The kinematic part is usually simply axiomatized by Newtonian space (a vector space $\Bbb R \times \Bbb R^3$ with an inner product on $\Bbb R^3$ and so forth), although you can model it as a manifold using the Newton-Cartan theory. The dynamic can then be done a variety of ways, either using the Newton equation directly, or via Lagrangian mechanics (a bundle approach is sometimes used for this as well), or Hamiltonian. You might want to check Arnold for more fun details on the topic. Nothing too complicated although regularity conditions are important to specify to avoid weird edge cases such as Norton's dome or the space invader. I also can't fail to mention the really stupid geometric axiomatization, which is absolutely not fit for any calculations but has the merit to exist.
EM, and by extension gauge theory in general, is done by the formalism of principal connections. You can find out more informations about it for instance in Topology, geometry and gauge fields.
Thermodynamics can be axiomatized using the headache-inducing contact manifolds, in a hellish formalism called geometrothermodynamics.
Those are about all the fields that have a really formal axiomatization that I can think of, but there are probably others.
