The orbit structure is extremely simple. If $p=2$, there is one orbit (the isometry group of a Hilbert space acts transitively), whereas for $p\neq 2$ there are exactly $2$ orbits: the (classes of) functions that do not vanish on a set of positive measure and its complement, the functions that do vanish on a set of positive measure.

This is certainly well-known, but I do not remember where this is written (I thought that it might in a paper by Ferenczi and Rosendal, but I could not locate it). Let me provide the proof in the interesting case $p\neq 2$.

Let me first prove that if two functions $f$ and $g$ in $S_X$ do not vanish, then they are in the same orbit. The two probability spaces $([0,1], |f|^p d\lambda)$ and $([0,1], |g|^p d\lambda)$ (where $\lambda$ is the Lebesgue measure) are standard complete atomless probability spaces. They are therefore isomorphic. This implies that there is a bimeasurable bijection $\phi \colon [0,1] \to [0,1]$ which sends $|f|^p d\lambda$ to $|g|^p d\lambda$. The three maps

- $h \in L^p(d\lambda) \mapsto \frac{h}{f} \in L^p(|f|^p d\lambda)$,
- $h \in L^p(|f|^p d\lambda) \mapsto h \circ \phi^{-1} \in L^p(|g|^p d\lambda)$,
- $h \in L^p(|g|^p d\lambda) \mapsto gh \in L^p(d\lambda)$.

are all surjective isometries (here we use that $|f|>0$ and $|g|>0$ a.s.). Their composition $h \mapsto g \frac{h \circ \phi^{-1}}{f \circ \phi^{-1}}$ is therefore a linear isometry which maps $f$ to $g$.

If $f$ and $g$ both vanish (say on $A$ and $B$ respectively), we can find a linear isometry from $L^p(A,d\lambda)$ onto $L^p(B,d\lambda)$ and combine it with a linear isometry $L^p([0,1]\setminus A) \to L^p([0,1]\setminus B)$ given by the previous case to find a isometry of $L^p([0,1])$ that maps $f$ to $g$.

The converse (that a function that almost surely does not vanish cannot be in the orbit of a function that vanishes) follows from the Banach-Lamperti theorem, which characterizes the linear isometries of $L^p([0,1])$: all such isometries are of the form
$$ Uf(x) =\omega(x) f(T^{-1}(x))$$
for a measurable bijection $T \colon [0,1] \to [0,1]$ that preserves the class of the Lebesgue measure $\lambda$, and a function $\omega\colon[0,1] \to \mathbf{C}^*$ satisfying $|\omega(x)|^p = \frac{d T_*\lambda}{d\lambda}(x)$.