Instantons are solutions of the classical equation of motions in Euclidean, i.e. imaginary, time, not in Lorentzian, i.e. real, time.

If you are a classical physicist, you look at solutions of the classical equations of motion in real time and so you don't see instantons.

If you are a quantum physicist, you compute partition functions or more generally correlation functions by path integral over all the possible classical configurations (not necessarily solutions of the classical equation of motions) weighted by $e^{iS/g}$ where $S$ is the classical action in real time and where $g$ is some coupling constant which is assumed to be small. In the limit $g \rightarrow 0$, stationary phase approximation tells you that the path integral is dominated by a classical equation of motion, and you can compute corrections in powers of $g$ to the classical contribution: it is usual perturbative physics. But sometimes, perturbative physics do not capture the full physics. For example, some transitions are classically impossible and so there will be no corresponding classical solution, and so the above approximation will give zero as transition amplitude, whereas the true amplitude is maybe not zero. Typical examples are tunneling processes between configurations separated by a potential which is too high to be classically passed. To study this kind of processes, it is useful to do a Wick rotation, i.e. to go from real time to imaginary time, and then to recover results in real time by analytical continuation. The integrand in the path integral in imaginary time is now $e^{-S^{eucl}/g}$ where $S^{eucl}$ is the Euclidean action. Saddle point approximation tells you that this path integral is dominated by stationnary points of $S^{eucl}$, i.e. by solutions of the classical equations of motion in imaginary time, which are classically called instantons when non-trivial. The real time interpretation of instantons is as computing leading order terms for tunnelling effects. The contribution of an instanton is of the form $e^{-S^{eucl}_I /g}$ where $S^{eucl}_I$ is the Euclidean action of the instanton, and so cannot be seen in a power series expansion is $g \rightarrow 0$: ther are non-perturbative effects in $g$.

In a field theoretic context, what I have written above as to treated carefully because of possible quantum renormalizations of $g$. In particular, the instanton approximation can become useless when the effective coupling becomes strong (as it happens in quantum Yang-Mills theory).

In general, instantons give leading terms in a saddle path approximation to a Euclidean path integral. In general, they are not the exact answer: they are quantum perturbative effects around instantons, effective interactions between instantons... But the miracle of supersymmetric theories is that in such case the path integrals localize over supersymmetric configurations. If there is enough supersymmetry, path integrals localize to finite dimensional integrals over finite dimensional moduli spaces of supersymmetric instantons.

In non-supersymmetric Yang-Mills theory, the path integral is over the infinite dimensional space $\mathcal{A}/\mathcal{G}$ over gauge equivalence classes of connections. But in 4d Yang-Mills theory with $\mathcal{N}=2$ supersymmetry, this path integral localizes over the finite dimensional moduli spaces $\mathcal{M}_{inst}$ of instantons (it is the original physics interpretation of Donaldson invariants given by Witten).

If you are a mathematician, I don't think that the distinction classical/quantum really makes sense.