Is there a good way to understand the free loop space of a sphere?
Stably the free loop space of the suspension of a connected space splits up, just as the based loop space does. Just as $\Omega\Sigma X$ is stably the wedge of the smash products $X^{\wedge n}$, $L\Sigma X$ is stably the wedge of $S^1_+\wedge_{C_n}X^{\wedge n}$. Here $C_n$ is a cyclic group of order $n$ acting freely on $S^1$ and permuting the factors in the smash product. This is even correct $S^1$-equivariantly in a weak sense.
EDIT Here is an answer more directly relevant to the question:
Suppose $X\sim BG$. Even if the group $G$ is not discrete, we still have the equivalence that Qiaochu mentioned in a comment, between $LX$ and the homotopy orbit space for the conjugation action of $G$ on $G$. And this in turn is equivalent to the two-sided bar construction for $G$ acting on itself on both sides, i.e. the cyclic bar construction $N^{cyc}G$. And if we have another topological monoid $M$ equivalent to $G$, for example the Moore loops on $X$, or the James construction $JY$ if $X=\Sigma Y$, then $N^{cyc}M$ is another model for $LX$.
Furthermore, there is a nice way of thinking of (the realization of the simplicial space) $N^{cyc}M$, which informally goes like this: A point is given by a finite subset $T$ of $S^1$ together with a labeling: a function $T\to M$. The set $T$ and the labels can move. If $T$ moves in such a way that several points come together, the label of the new point is the product of the old labels. If a label becomes $1\in M$ then the point may be deleted from $T$.
This is even equivariantly correct regarding the $S^1$-action, in some sense (not good for fixed-point spaces of the whole group, but OK for finite subgroups).
When $X=S^{n}$ and $M=JS^{n-1}$, this gives a pretty good equivariant cell structure for $LS^n$.
The free loop space $LM$ forms a fibre bundle over $M$, by evaluating a loop at $1$, with fibre $\Omega M$. So, in theory, one "knows" a lot if one knows the twisting involved when building this bundle/fibration. There are several computational tools that help you to compute Betti numbers of $LM$ etc (rational homotopy theory, Hochschild homology etc.) but since you ask for CW complexes for small values of $n$, here is an attempt :
For any group $G$, it is clear that $LG\cong G\times\Omega G$.This is not an isomorphism as groups! In particular, when $n=1$ or $3$, we have a description of $LS^1\sim S^1\times\mathbb{Z}$ and $LS^3$. Even spheres and higher odd dimensional spheres ($5$, $9$ and higher) do not fall in this list. The Betti numbers of $LS^n$ are computable and these are bounded (either $0$, $1$ or $2$). This should give you a (rational) model for the $n$th skeleton of $LM$. And, in a non-rigorous sense which is meaningful in the context of algebraic structures, $LS^\textrm{odd}$ "behaves" like $S^\textrm{odd}\times\Omega S^\textrm{odd}$.
There is a relationship between $\Omega M$ and $LM$ via homological algebra for simply connected manifolds $M$. The Hochschild homology of the chains on $\Omega M$ is isomorphic to the cohomology ring of $LM$. This is perhaps as precise as it gets; any simple relationship that you may want needs to account for the fact that $LM$ has a circle action while $\Omega M$ doesn't.
The next best thing to the knowledge of a CW-structure is probably the knowledge of the integral homology (not just the Betti numbers). Perhaps Ziller was the first to compute it in The Free Loop Space of Global Symmetric Spaces. Note the following:
In the case of $n>2$, where $LS^n$ is simply-connected, this gives the number of cells of a minimal CW-structure (see Hatcher's Algebraic Topology).
The behavior in the case $n$ even and $n$ odd is very different. The difference might be seen as a differential in a Serre spectral sequence, but I like to view it as coming from the hairy ball theorem that we cannot find a non-vanishing vector field on an even-dimensional sphere.
The latter connection becomes clear if we look at explicit generators of the integral homology. I have done this in Section 5.2.1 of my paper Spectral Sequences in String Topology (I apologize for the self-advertisement).