The core of the Noether theorem in all contexts where it arises is surprisingly elementary!
From a very general point of view, one considers the following structure.
(i) A set of "states" $x\in \Omega$,
(ii) A one-parameter group of transformations of the states $\phi_u : \Omega\to \Omega$, where $u\in \mathbb{R}$.
These transformations are requested to satisfy by definition $$\phi_t\circ \phi_u = \phi_{t+u}\:, \quad \phi_{-u}= (\phi_u)^{-1}\:, \quad \phi_0 = \text{id}\tag{0}\:.$$
(iii) A preferred special one-parameter group of transformations
$$E_t : \Omega \to \Omega $$
representing the time evolution (the dynamics) of the physical system whose states are in $\Omega$.
The general physical interpretation is clear. $\phi_u$ represents a continuous transformation of the states $x\in \Omega$ which is additive in the parameter $u$ and is always reversible. Think of the group of rotations of an angle $u$ around a given axis or the set of translations of a length along a given direction.
A continuous dynamical symmetry is a one-parameter group of transformations that commutes with the time evolution,
$$E_t \circ \phi_u = \phi_u \circ E_t \quad \forall u,t \in \mathbb{R}\:.\tag{1}$$
The meaning of $(1)$ is that if I consider the evolution of a state
$$x_t = E_t(x)$$
and I perform a symmetry transformation at each time
$$ \phi_u(x_t)\:,$$
then
the resulting time-parametrized curve of states is still a possible evolution with respect to the said dynamics
$$\phi_u(x_t) = E_t(\phi_u(x))\:.$$
These features are shared by the theory of dynamical systems, Lagrangian mechanics, Hamiltonian mechanics, Quantum Mechanics, and general Quantum Theory including QFT.
The difference is the mathematical nature of the space $\Omega$ and some continuity/differentiability properties of the map $\mathbb{R} \ni u \mapsto \phi_u$, whose specific nature depends on the context.
The crucial observation is that once assumed these quite natural properties,
the one-parameter group structure $(0)$ provides a precise meaning of
$$X := \frac{\mathrm{d}}{\mathrm{d}u}\Biggr|_{u=0} \phi_u$$
and, exactly as for the standard exponential maps which satisfy $(0)$, one has (for us, it is just a pictorial notation)
$$\phi_u = e^{uX}\:.$$
$X$ is the generator of the continuous symmetry.
In quantum theory, $X$ (more precisely $iX$) is a self adjoint operator and hence a quantum observable,
in dynamical system theory and Lagrangian mechanics, $X$ is a vector field,
in Hamiltonian mechanics $X$ --written as $X_f$ -- is an Hamiltonian vector field associated to some function $f$.
$X$ (or $iX$, or $f$) has another meaning, the one of observable.
However, it is worth stressing that this interpretation is delicate and strictly depends on the used formalism and on the mathematical nature of the space $\Omega$ (for instance, in real quantum mechanics, the said interpretation of $X$ in terms of an associated quantum observable is not possible in general).
Now notice that, for a fixed $t\in \mathbb{R}$,
$$u \mapsto E_t\circ e^{uX} \circ E^{-1}_t =: \phi^{(t)}_u$$
still satisfies $(0)$ as it immediately follows per direct inspection. Therefore, it can be written as
$$E_t\circ e^{uX} \circ E^{-1}_t = e^{uX_t}\tag{3}$$
for some time-depending generator $X_t$.
We, therefore, have a time-parametrized curve of generators
$$\mathbb{R} \ni t \mapsto X_t\:.$$
The physical meaning of $X_t$ is the observable (associated to) $X$ temporally translated to the time $t$.
That interpretation can be grasped from the equivalent form of $(3)$
$$E_t \circ e^{uX} = e^{uX_t} \circ E_t \tag{4}.$$
The similar curve
$$\mathbb{R} \ni t \mapsto X_{-t}$$
has the meaning of the time evolution of the observable (associated to) $X$.
One can check that this is, in fact, the meaning of that curve in the various areas of mathematical physics I introduced above. In quantum mechanics, $X_t$ is nothing but the Heisenberg evolution of $X$.
Noether Theorem.
$\{e^{uX}\}_{u\in \mathbb R}$ is a dynamical symmetry for $\{E_t\}_{t\in \mathbb R}$ if and only if $X=X_t$ for all $t\in \mathbb R$.
PROOF.
The symmetry condition $(1)$ for $\phi_t = e^{tX}$ can be equivalently rewritten as
$E_t \circ e^{uX} \circ E^{-1}_t = e^{uX}$. That is, according to $(3)$:
$e^{uX_t} = e^{uX}$. Taking the $u$-derivative at $u=0$ we have $X_t=X$ for all $t\in \mathbb R$. Proceeding backwardly $X_t=X$ for all $t\in \mathbb R$ implies
$(1)$ for $\phi_t = e^{tX}$. QED
Since $E_t$ commutes with itself, we have an immediate corollary.
Corollary. The generator $H$ of the dynamical evolution
$$E_t = e^{tH}$$
is a constant of motion.
That is mathematics. The existence of specific groups of symmetries is a matter of physics.
It is usually assumed that the dynamics of an isolated physical system is invariant under a Lie group of transformations.
In classical mechanics (in its various formulations), that group is Galileo's one. In special relativity, that group is Poincaré's one. The same happens in the corresponding quantum formulations.
Every Lie group of dimension $n$ admits $n$ one-parameter subgroups. Associated with each of them, there is a corresponding conserved quantity when these subgroups act on a physical system, according to the above discussion. Time evolution is one of these subgroups.
The two aforementioned groups have dimension $10$, and, thus, there are $10$ (scalar) conserved quantities. Actually, $3$ quantities (associated with Galilean boosts and Lorentzian boosts) have a more complex nature and require a bit more sophisticated approach, which I will not discuss here; the remaining ones are well known: energy (time evolution), three components of the total momentum (translations along the three axes), three components of the angular momentum (rotations around the three axes).
