It is a difficult problem to determine all the characteristic subgroups of a non-abelian $p$-group. But, then, I would seek for as many characteristic subgroups as we can, for small order $p$-groups.
For a non-abelian $p$-group, the most natural list will start from
$Z(G)$, $[G,G]$, $\mho_n(G)=\langle g^{p^n}\colon g\in G\rangle$, $\Omega_n(G)=\langle g\colon g^{p^n}=1\rangle.$
I would call these type of characteristic subgroups as primary characteristic subgroups, since the next characteristic subgroups are obtained from these one. The following are some other characteristic subgroups obtained from primary ones:
One may form $HK$, $H\cap K$, $[H,K]$ where $H,K$ are characteristic.
For $H$ a characteristic subgroup, take $G/H$ and pull back its characteristic subgroup in $G$.
Question: Are there some other primary characteristic subgroups for a non-abelian $p$-group?
Note: In $p$-groups, Frattini is product of two primary characteristic subgroups: $[G,G]$ and $\mho_1(G)$.