All our perceptions of the world consist in the information
(like files in computers, even if the correspondence between
natural and digital types of information is fuzzy) that our senses
(nerves) give us. They do not directly reveal (at least not in an
obvious way) any ontological aspects, i.e. "real nature" of things
(or aspects of the world, such as time), that may be causes, or
sources, of our perceptions. Without trying to directly discuss
any ontology, the work of science is to develop our understanding
of the world as follows:
This structure of observable information, once understood and formulated in the form of theories, even while remaining on a scientific level (giving the practical shape of things, what we can work with in practice without entering any ontological debate) is already a rich, fascinating, and often sufficient understanding of the world.
See also what are mathematics and mathematical theories, and some longer comments on the nature of science and reason.
Among sciences, Physics focuses on fields or aspects of the
world that are especially "elementary", which happened to be
successfully described by mathematical theories (thus rigorously
clear about their consequences, without risk of being relative to
anyone's fanciful interpretation). Aspects that way "elementary",
include
Curved spacetime
(Lorentzian manifold with signature (3, 1)) 

Minkowski spacetime (Special Relativity) 
Classical spacetime
(absolute time)
with Equivalence principle of General Relativity 
Galilean
spacetime (relative speed) 

Aristotelian
spacetime absolute speeds: some things stay fixed (on the ground), others move 

Euclidean
(3dimensional)
space (no movement) 
The two fundamental ones from which all others are
approximations, are the Abstract Quantum Theory and the Theory of
Quantum States and Measurements. We cannot really define an order
between them. The Abstract Quantum Theory generally consists in
operating with Hilbert spaces (though this definition is linked to
the role of time; this theory may be expressed in other ways
depending on its articulation with spacetime). Its laws are
timereversible (where reversing time goes with taking conjugation
of complex numbers). But it is a mere abstract mathematical theory
that does deal with any effective realities.
The Theory of Quantum States and Measurements, works with positive
Hermitian forms on these Hilbert spaces. Thus its mathematical
content is simply built over the objects of Abstract Quantum
Theory, but in a time dissymmetric way, and with its special,
metaphysical rather than strictly mathematical concepts of "what
is real" (states of physical systems, and what is the form of
produce experimental predictions).
The only proper frameworks where to express fundamental specific
laws (the different types of elementary particles and fields), are
the Abstract Quantum Theory and the Least Action Principle (while
the specific laws in others frameworks are consequences or
phenomenological versions of these fundamental specific laws). We
may roughly consider the Least Action Principle as an
approximation of the Abstract Quantum Theory, however most of the
specific fundamental laws can in a first step be defined in the
framework of the Least Action Principle, after which a more or
less standard mathematical tool called quantization, converts them
into a specific fundamental law in the framework of Abstract
Quantum Theory.
Conservation Laws are exact consequences of the Least Action
Principle (thus not approximations, but still less fundamental).
They remain valid from a macroscopic viewpoint, where the quantum
undeterminations of their object vanishes, but while we lose trace
of the fundamental (microscopic) processes underlying the
considered processes. For example the phenomenon of friction
satisfies the conservation laws but cannot be analyzed in terms of
the least action principle.
Symplectic Chaos and Dissipative Chaos are but probabilistic
studies of the consequences of classical physics. The symplectic
case comes from Least Action Principle, while the dissipative case
comes from the mere conservations laws completed by classical
thermodynamics and other phenomenological specific laws roughly
determining behaviors in short time intervals.
The double choice of a type of spacetime and a principle of
mechanics, first constitutes itself a theoretical framework that
can be given its own name. Then, it can allow for (include) for
several types of particles or fields (but particles and fields are
the same thing for quantum field theory) described by their
respective specific laws. So let us put them all in a big table.
Columns represent principles of mechanics ; lines represent types
of spacetime.
Each cell first contains the name of the framework defined by this
double choice, then specific laws for particles and fields that
can interestingly fit there. Let us just keep here both principles
of mechanics where fundamental specific laws are expressed:
Abstract Quantum Theory 
Least Action Principle 

Curved spacetime 
Quantum gravity (more
tricky than a curved spacetime because of quantum
indeterminations) TOE (theory of everything, yet to be discovered) 
Electromagnetism in general relativity. As simple
particular cases we have cosmology,
black holes, gravitational waves... 
Minkowski
spacetime 
Quantum Field Theory
Inside it : specification of the Standard Model, including: QCD = Quantum Chromodynamics (nuclear force); Electroweak interaction, that approximates into QED (quantum electrodynamics = the quantized version of electromagnetism) 
Relativistic Mechanics
Electromagnetism Nuclear reactions 
Classical with Equivalence principle  Classical gravitation and
cosmology (allows for a universe with a density of matter
that does not decrease for large distances) 

Galilean spacetime 
Schrodinger equation for a
system of particles 
Classical mechanics; usual
expression of Newton's law of gravitation (not requiring the
equivalence principle but requiring vanishing of density at
large distances); electricity (dynamics of electric charges,
neglecting magnetic induction); magnets and electricity with
magnetic induction (neglecting capacitors) 
Aristotelian spacetime  Chemistry 
Mechanics of matter,
acoustics, electromagnetism in matter (that may be
well explained in that article, though I did not read
it in details) 
Euclidean space 
Equilibrium (stable or
unstable cases  the role of action is played by potential
energy) : Electrostatics, magnetostatics, geometric optics. 
Thanks to the hard work accomplished by physicists, we have now enough theories to form a clear pack of complementary description and prediction tools addressing (well predicting) seemingly all accessible physical processes without conflict (not contradicting each other nor disturbing each other in nonunderstood ways in any practically accessible field of observation). Namely, almost every specific accessible physical phenomenon can be either properly described by some physical theory (i.e. is in the range of its good approximate validity, without visible margin of errors), or split into several phases of process that can each be described by some theory, so that a succession or combination of computations for each phase of process from its relevant theory, can produce good effective predictions for the global phenomenon.
But despite this success, we might still feel not fully satisfied
as we have not one theory but different theories for different
ranges of observable phenomena (or phases of computations of
phenomena), and we do not currently know how to unify them (or
rather, the most fundamental among known ones, from which others
come as approximations) into a single fully coherent theory, that
would make exactly welldefined predictions (thus, predictions on
any observation "possible in principle"), and thus would more
deeply explain everything as deduced from a clear common
foundation.
The connections between quantum field theory and gravity by other
theories (namely, relativistic mechanics and the Least Action
Principle), provide hints in the search for possible theories
unifying them (while the issue of quantum measurement and
thermodynamics is usually ignored by necessity, by lack of
reasonable possibilities to dig into it), but anyway this search
is extremely hard, involving very high mathematical works where it
is far from clear on what conceptual basis might things be
coherently defined (anyway, even if found, a unification theory
will probably remain very hard to understand, quite harder than
current theories).
In fact this problem of currently having a mere "pack of
prediction tools" with a limited but satisfying accuracy (enough
well defined for what our limited tools of observation could check
in practice) rather than a mathematically welldefined and
coherent theory, is not specific to the problem of uniting several
existing theories, but is already an intrinsic defect of one of
these known fundamental "theories", namely the Standard Model.
Indeed, this "theory" is full of hard mathematical concepts but
only forming a sort of approximate expression of a theory, not a
fully welldefined mathematical theory, despite the huge play of
internal reformulations that were applied. Some extreme physical
phenomena (very hard to detect: first fractions of second of the
big bang, highenergy collisions, details of dark matter...),
still escape predictability from the Standard model, and would
require to find either a more coherent expression of it, or some
more fundamental theory (may it be or not the ultimate one).
But these "extreme phenomena" still do not explicitly touch the
problem of quantum gravity. If considered just for itself, the
problem of finding a "ultimate theory" unifying gravity with
quantum theory (and more precisely with the Standard Model), would
be practically pointless (not effectively qualifiable as a
"scientific question"), as it would only concern much too "extreme
phenomena" (too far from any reasonable hope of ever touching
mankind's field of possible observations, even after millenia).
To most efficiently learn theoretical physics, we need to study
it in the form of mathematical theories. While our available
understanding abilities (structures of imagination) are the way
they are coming from our human nature, the optimal ways to use
them to understand the known concepts of theoretical physics
(structures of the world), are a matter of being convenient
(optimal) to the clear expression of the mathematical theories,
which may require to break the specific connections (dependence)
of our imagination with the world, usually given by our daily
direct perceptions (at least for things we happen to perceive at
all), or that may come from any ontological ideas we may have.
Such mathematically optimal ways to imagine things (the practical
level of knowledge which suffices in most cases), while in
principle closer to these things, may still remain as empty of
significance to any possible "true nature" (ontology) of these
things, as any other discussion. Example : discussions about time
in relativity theory.
It is possible to tell a lot of things about fundamental physics
with very little mathematical background, namely with the mere dimensional
analysis  even if this is just a sort of popularization
without genuine understanding of the theories.
More fundamental physics, especially special relativity
and quantum
physics, can be done just based on the understanding of
affine geometry and
vector spaces.
The mathematical formalism of tensors is necessary for
a clean general expression of most theories of fundamental
physics, including classical mechanics.
It is reputed "difficult", but in fact not only because it is
really difficult (somehow it is but not necessarily so much), but
also because the common way of introducing this formalism is not
as clear as it could and would deserve to be.
Thus ideally, a scientific curriculum that aims to teach physics
up to some good level, should give some priority to the
development of the algebraic concepts needed for the clean
introduction of tensors. It would not be quite hard for students
if it was well done (as compared to the current first year
mathematics curriculum); it just requires some changes, some new
concepts and ways to define things.
In parallel and before the achievement of that goal, some
theoretical physics can already be introduced. But for the general
optimization of the curriculum, it would be better to focus this
teaching specifically on those concepts that really don't need
tensors  so as to not pitifully waste efforts making up odd,
obscure mathematical tools just to express without any explicit
use of tensors, some concepts that would in fact need them.
Already the following concepts are example of such things made
made odd (obscure) by the try to express them without their
natural framework of tensors:
Also, quantum theory may be first introduced without tensors in
the simplest cases (as done in the pages linked above), but
require tensors as soon as we want to consider more complex cases.
It is wellknown that general relativity is normally expressed
with tensors, but there is still some way to simply express it
without tensors (but only in principle, in a way that does not
appear like "a formula" but as concepts to be carefully applied,
and only usable in practice for the simplest, most symmetric types
of applications such as cosmology and the Schwarzchild black hole)
Now, what are the theories and other useful mathematical concepts
for physics, that can be cleanly introduced before tensors, and in
which order ?
Here are they, and a possible order in which to put them.