(Edit made in view of a comment by Richard Lyons correcting an inaccuracy in my former answer)
A useful invariant of quadratic forms is the Witt ring $W$. The 1-dimensional form $x^2$ correspond to $1\in W$. The standard form, which I will denote $A_1$, satisfies $[A_1]=n\cdot 1 \in W$. It is easy to see that $[A_0]\in W$ is 0 if $n$ is even and it 1 if $n$ is odd. The order of 1 in $W$ is 2 if $p$ is 1 mod 4 and it is 4 if $p$ is 3 mod 4. It follows that $A_0$ an d$A_1$ are equivalent if $p$ is 1 mod 4 or if $p$ is 3 mod 4 and $n$ is 0 or 1 mod 4. In case both $p$ and $n$ are 3 mod 4, $A_0$ and $A_1$ are not equivalent, but differ (up to equivalence) by a (non-square) scalar multiplication. In all of these cases $\text{SO}(A_0,\mathbb{F}_p)$ and $\text{SO}(\mathbb{F}_p)$ are conjugated in $\text{GL}(\mathbb{F}_p)$. In the remining case, where $p$ is 3 mod 4 and $n$ is 2 mod 4, these groups are non-isomorphic and their "subgroup structure" is different. For example, the split group $\text{SO}(A_0,\mathbb{F}_p)$ contains a copy of $(F_p^*)^{n/2}$, while $\text{SO}(\mathbb{F}_p)$ does not.
I am quite confident that a subgroup of minimal index in either group is the stabilizer $Q$ of an isotropic line in $\mathbb{F}_p^n$. It is certainly minimal among algebraic subgroups. The orbit of an isotropic line in the projective space $\mathbb{P}^{n-1}(\mathbb{F}_p)$ is the zero locus of the corresponding quadratic form, thus it is a variety of dimension $n-2$. It follows that $[G:Q]$ is $\sim p^{n-2}$.
$Q$ is a one of the maximal parabolic subgroups - these are the stabilizers of isotropic subspaces of $\mathbb{F}_p^n$ (of dimension $1,\dots,[n/2]$ for the split form $A_0$). The others are also of relatively small index, but bigger then that of $Q$. Other subgroups of small index are reductive subgroups, such as the stabilizer of an anisotropic vector in $\mathbb{F}_p^n$, which is of codimension $n-1$, thus of index $\sim p^{n-1}$. Again, you can take stabilizers of subspace of higher dimension, but you will pay for this with a bigger index.