The first unstable homotopy group of $Sp(n)$

The answer appears to be in the paper Homotopy groups of symplectic groups by Mimura and Toda. They claim the calculation was already in a paper of Harris, but that was stated in terms of a symmetric space and it's not immediately obvious to me how to translate into information about the groups.

They state that the group is $\mathbb Z_{(2n+1)!}$ if $n$ is even and $\mathbb Z_{(2n+1)! \cdot 2}$ if $n$ is odd, which agrees with your data.

The first unstable homotopy groups of $SO(n)$ are actually 8-periodic (except for some junk at the beginning). Some more unstable homotopy groups of $SO(n)$ can be found in:

• M. Kervaire. Some nonstable homotopy groups of Lie groups. Illinois J. Math. 4 (1960), 161-169. (link to journal website)

The 8-periodicity for the orthogonal group comes about as follows: the relevant piece of the stabilization sequence is $$ \pi_n S^n\to \pi_{n-1}SO(n)\to \pi_{n-1}SO(n+1)\to 0. $$ The unstable homotopy groups $\pi_{n-1}SO(n)$ are then direct sums of the stable stuff from $\pi_{n-1}SO(\infty)$ plus a cyclic quotient of $\pi_n S^n\cong \mathbb{Z}$. The 8-periodicity effectively comes from the stable summand (check the list of homotopy groups of the infinite orthogonal group). The cyclic quotient of $\pi_n S^n$ is only 2-periodic, alternating between $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$; I think this basically comes from the corresponding Euler class of the sphere alternating between 2 and 0.

The description of the unstable homotopy of the symplectic groups given in Will Sawin's answer can also be found in

• B. Harris. Some calculations of homotopy groups of symmetric spaces. Trans. Amer. Math. Soc. 106 (1963), 174-184. (link to journal website)