There is one naive Riemannian metric on $C^\infty(S^1,M)$, namely
$$
G_c(h,k) = \int_{S^1} g(h,k) d\theta
$$
where $c$ is a smooth curve, $h,k$ are vector fields along the curve, and $d\theta$ is a fixed measure on $S^1$. Here the points $c(\theta)$ are freely (decoupled from each other) following geodesics in $M$,
curvature is just induced by curvature on $M$.
So a closed geodesic for this metric is just an $S^1$ parameterized family of closed geodesic in $M$.
This metric is not invariant under the action of $Diff(S^1)$ from the right.

If you want metrics which are invariant under reparametrization ($Diff(S^1)$-invariant),
you have the metric
$$
G^0_c(h,h) = \int_{S^1} g(h,k) ds
$$
where $s=s_c = \|c'(\theta)\|d\theta$ is the arc length measure. this is a Riemannian metric only on the open submanifold $Imm(S^1,M)$ of all immersions, since the arc length measure can become degenerate. But this metric has everywhere vanishing geodesic distance, see

Peter W. Michor; David Mumford: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8 (2006), 1-48. (pdf)

Peter W. Michor; David Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Math. 10 (2005), 217--245. (pdf)

In order to remedy the vanishing geodesic distance one has to go to higher Sobolev Riemannian metrics or add functions of arc length or curvature. A quite comprehensive overview article on all this is:

- Martin Bauer, Martins Bruveris, Peter W. Michor: Overview of the Geometries of Shape Spaces and Diffeomorphism Groups. Journal of Mathematical Imaging and Vision, 50, 1-2, 60-97, 2014. (here)

You ask about closed geodesics. Before that question it would be good to know that the metric on the space of immersions is geodesically complete (Hopf-Rinov falls apart in infinite dimensions). Results about that are still scarce: The Sobolev $H^2$ metric with coercive term on $Imm(S^1, \mathbb R^k)$ is geodesically complete.