Small-index subgroups of SL(3,Z)
To fill in the comments, there are basically two serious issues involved.
1) You want to know that every subgroup of finite index in $\mathrm{SL}(3,\mathbb{Z})$ contains some congruence kernel: the kernel of the natural reduction homomorphism induced by $\mathbb{Z} \rightarrow \mathbb{Z}/n\mathbb{Z}$. In other words, the original group satisfies the congruence subgroup property (as HW comments). This is already quite nontrivial to prove and evolved from work of Bass-Lazard-Serre here followed by more definitive work by Bass-Milnor-Serre here. (There is some exposition in the last part of my 1980 Springer Lecture Notes 789.)
2) Now you have to pin down the maximal subgroups of a typical finite group $\mathrm{SL}(3,n)$ and pull these back to the big group in order to sort out which give you subgroups of minimal index there. Maximal subgroups of the finite groups in question have been extensively studied, especially over finite fields, but in this case some trial-and-error leads pretty quickly to the number 7, as indicated by Aakumadula. This results from the first possibility $n=2$, where you get the simple nonabelian group $\mathrm{SL}(3,2)$ of order $ 8 \cdot 3 \cdot 7$, from the general formula for a prime $p$ given by $p^3 (p^2-1)(p^3-1)$. Here there is a maximal parabolic subgroup of index 7.
The search for maximal subgroups is itself a major project, for which I don't have all relevant references at hand. But over a field some basic insight comes from algebraic groups, where you see that a maximal (closed) proper subgroup will be either parabolic or reductive. I'm not sure in the special case at hand how hard it is to pin down the maximal subgroups of the various finite groups directly.
ADDED: As pointed out in comments, it's possible here to rule out subgroups of index 2 or 3 by combining the known commutator group result with a study of actions of the big group on left cosets of a hypothetical subgroup. But a systematic study (especially for other Chevalley-type groups) probably requires something like the program I've sketched above. Given the congruence subgroup property, the study of subgroups of finite index then reduces quickly to the study of finite groups of the same type over finite rings and then prime fields. In your case, this helps to provide a conceptual answer to such questions as: Why does $\mathrm{SL}(3, \mathbb{Z})$ have a subgroup of index 7?
There is a vast literature by now on maximal subgroups of known finite simple groups, by people including Aschbacher, Kleidman, Liebeck, Scott, Seitz, .... An older brief survey by Kleidman-Liebeck, with an impressive bibliography, appeared in Geom. Dedicata 25 (1988). For your specific group there are very old results, but the general approach is more unified even though intricate.
In order to answer the question we need a finite presentation of ${\rm SL}(3,\mathbb{Z})$ and a general method to find all subgroups of index $\leq n$ of a finitely presented group:
A finite presentation for ${\rm SL}(3,\mathbb{Z})$ can be found for example in Theorem 2 in
Marston Conder, Edmund Robertson, Peter Williams: Presentations for 3-dimensional special linear groups over integer rings, Proc. Amer. Math. Soc. 115 (1992), no. 1, 19-26. http://www.ams.org/journals/proc/1992-115-01/S0002-9939-1992-1079696-5/S0002-9939-1992-1079696-5.pdf.
The finite presentation given in this paper is $$ {\rm SL}(3,\mathbb{Z}) \cong \left< x, y, z \ | \ x^3 = y^3 = z^2 = (xz)^3 = (yz)^3 = (x^{-1}zxy)^2 = (y^{-1}zyx)^2 = (xy)^6 = 1 \right> $$ on the generators $$ x \ = \ \left( \begin{array}{rrr} 1 & 0 & 1 \\\ 0 & -1 & -1 \\\ 0 & 1 & 0 \end{array} \right), \ \ y \ = \ \left( \begin{array}{rrr} 0 & 1 & 0 \\\ 0 & 0 & 1 \\\ 1 & 0 & 0 \end{array} \right), \ \ z \ = \ \left( \begin{array}{rrr} 0 & 1 & 0 \\\ 1 & 0 & 0 \\\ -1 & -1 & -1 \end{array} \right). $$
A general method to find all subgroups of index $\leq n$ of a finitely presented group is the so-called low index subgroups procedure. This algorithm is described in Section 5.4 in
Derek F. Holt, Bettina Eick, and Eamonn A. O'Brien, Handbook of computational group theory, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall / CRC, Boca Raton, FL, 2005. MR 2129747 (2006f:20001).
For an online resource, see e.g.
Marston Conder: Applications and adaptations of the low index subgroups procedure, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.107.5164.
The low index subgroups procedure is implemented in the GAP computer algebra system (cf. http://www.gap-system.org). Hence all we need to do is to enter the presentation of ${\rm SL}(3,\mathbb{Z})$ taken from the above paper into GAP ...
gap> F := FreeGroup("x","y","z");;
gap> AssignGeneratorVariables(F);
#I Assigned the global variables [ x, y, z ]
gap> G := F/[x^3,y^3,z^2,(x*z)^3,(y*z)^3,(x^-1*z*x*y)^2,(y^-1*z*y*x)^2,(x*y)^6];
<fp group on the generators [ x, y, z ]>
... and to run the algorithm on it:
gap> sub := LowIndexSubgroupsFpGroup(G,7);;
gap> List(sub,H->Index(G,H));
[ 1, 7, 7 ]
gap> gens := List(sub,GeneratorsOfGroup);
[ [ x, y, z ], [ x, z, y*z*y^-1, (y*x)^2*y ],
[ x, y*x^-1*z^-1, y^-1*z*y, z*y^-1*x*y, y^-1*x*y*x^-1*y ] ]
This tells us that the smallest index of a proper subgroup of ${\rm SL}(3,\mathbb{Z})$ is 7, and that there are 2 conjugacy classes of subgroups of index 7. Now it is straightforward to obtain generators for our subgroups in terms of matrices:
gap> x := [ [ 1, 0, 1 ], [ 0, -1, -1 ], [ 0, 1, 0 ] ];;
gap> y := [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ];;
gap> z := [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ -1, -1, -1 ] ];;
gap> List(gens[2],g->MappedWord(g,GeneratorsOfGroup(G),[x,y,z]));
[ [ [ 1, 0, 1 ], [ 0, -1, -1 ], [ 0, 1, 0 ] ],
[ [ 0, 1, 0 ], [ 1, 0, 0 ], [ -1, -1, -1 ] ],
[ [ 0, 0, 1 ], [ -1, -1, -1 ], [ 1, 0, 0 ] ],
[ [ -1, -1, -1 ], [ 0, 0, 1 ], [ 0, 1, -1 ] ] ]
gap> List(gens[3],g->MappedWord(g,GeneratorsOfGroup(G),[x,y,z]));
[ [ [ 1, 0, 1 ], [ 0, -1, -1 ], [ 0, 1, 0 ] ],
[ [ -1, -1, -1 ], [ 0, 1, 1 ], [ 0, 0, -1 ] ],
[ [ -1, -1, -1 ], [ 0, 0, 1 ], [ 0, 1, 0 ] ],
[ [ 1, 1, 0 ], [ 0, 0, 1 ], [ 0, -1, 0 ] ],
[ [ -1, 0, -1 ], [ 2, 1, 1 ], [ 0, -1, 0 ] ] ]
So our representatives for the conjugacy classes of subgroups of ${\rm SL}(3,\mathbb{Z})$ of index 7 are $$ G_{7,1} \ = \ \left< \left(\begin{array}{rrr} 1&0&1\\\ 0&-1&-1\\\ 0&1&0 \end{array}\right), \ \left(\begin{array}{rrr}% 0&1&0\\\ 1&0&0\\\ -1&-1&-1 \end{array}\right), \\ \left(\begin{array}{rrr}% 0&0&1\\\ -1&-1&-1\\\ 1&0&0 \end{array}\right), \ \left(\begin{array}{rrr}% -1&-1&-1\\\ 0&0&1\\\ 0&1&-1 \end{array}\right) \right> $$ and $$ G_{7,2} \ = \ \left< \left(\begin{array}{rrr}% 1&0&1\\\ 0&-1&-1\\\ 0&1&0 \end{array}\right), \ \left(\begin{array}{rrr}% -1&-1&-1\\\ 0&1&1\\\ 0&0&-1 \end{array}\right), \ \left(\begin{array}{rrr}% -1&-1&-1\\\ 0&0&1\\\ 0&1&0 \end{array}\right), \\ \left(\begin{array}{rrr}% 1&1&0\\\ 0&0&1\\\ 0&-1&0 \end{array}\right), \ \left(\begin{array}{rrr}% -1&0&-1\\\ 2&1&1\\\ 0&-1&0 \end{array}\right) \right>. $$ The computations above take just a few milliseconds. If one is willing to put in a minute or so, then one can go a bit further and compute representatives for the conjugacy classes of subgroups of ${\rm SL}(3,\mathbb{Z})$ of index $\leq 30$:
gap> sub := LowIndexSubgroupsFpGroup(G,30);;
gap> List(sub,H->Index(G,H));
[ 1, 8, 7, 28, 14, 13, 7, 13, 28, 26, 14, 26, 28, 28, 28, 24, 21 ]
So we have subgroups of indices 7, 8, 13, 14, 21, 24, 26 and 28, and there are no proper subgroups of other indices $\leq 30$. Generators of the subgroups in terms of our generators $x, y, z$ of ${\rm SL}(3,\mathbb{Z})$ can be determined easily as well:
gap> List(sub,GeneratorsOfGroup);
[ [ x, y, z ], [ x, y ], [ x, z, y*z*y^-1, (y*x)^2*y ],
[ x, z, y*z*y^-1, y*x*(y*x^-1)^2*y, (y*x)^2*(y^-1*x)^2*y^-1 ],
[ x, z, (y*x)^2*y ], [ x, y*x^-1*z^-1, y^-1*z*y, z*y^-1*x^-1*y ],
[ x, y*x^-1*z^-1, y^-1*z*y, z*y^-1*x*y, y^-1*x*y*x^-1*y ],
[ x, y*x^-1*z^-1, y^-1*z*y, z*y^-1*x*y, y^-1*(x*y)^2*x^-1*y^-1*x^-1*y ],
[ x, y*x^-1*z^-1, y^-1*z*y, z*y^-1*x*y,
y^-1*(x*y)^2*x^-1*y^-1*x*y*x*y^-1*x^-1*y,
y^-1*(x*y)^2*x^-1*y*x*y^-1*x*y*x^-1*y^-1*x^-1*y ],
[ x, y*x^-1*z^-1, z*y^-1*x^-1*y ],
[ x, y*z*x^-1*y^-1, z*y*x*z^-1, (y*x)^2*y, y^-1*x*y*x^-1*y ],
[ x, y*z*x^-1*y^-1, z*y*x*z^-1, z*y^-1*x*y, y^-1*(x*y)^2*x^-1*y^-1*x^-1*y ],
[ x, y^-1*z*y, (y*x)^2*y, y^-1*x*y*x^-1*y ],
[ x, y^-1*z*y, (y*x)^2*y, z*y^-1*(x^-1*y)^2 ],
[ x, y^-1*z*y, y*x^-1*z*y^-1*x^-1*y^-1, y^-1*x*y*z^-1*x^-1*y^-1 ],
[ y*x^-1, y^-1*x ], [ z, x*z*x^-1, y*z*y^-1, (y*x)^3 ] ]
You don't need the congruence subgroup property to see that this group has no proper subgroups of index less than $7$, in other words that it has no nontrivial homomorphism to $S_6$.
Nor do you need a presentation of the group; all you need is that it is generated by six elements satisfying certain relations, and a little patience.
Let $x_{i,j}$ be the matrix with $1$ on the diagonal, $1$ in the $(i,j)$, spot, and $0$ otherwise. It is not hard to see, by row reduction, that $SL(3,\mathbb Z)$ is generated by these elements. When $i$, $j$, and $k$ are distinct then $x_{i,j}$ is the commutator of $x_{i,k}$ and $x_{k,j}$, and also $x_{i,j}$ commutes with $x_{i,k}$ and with $x_{k,j}$.
Now suppose for contradiction that $S_6$ has elements $x_{i,j}$ satisfying these same relations and not all equal to the identity. No $x_{i,j}$ can be the identity because then by taking commutators they all would be the identity. Being a commutator, $x_{i,j}$ must belong to $A_6$. It is the commutator of $x_{i,k}$ and $x_{k,j}$, two elements of $A_6$ that commute with it. Looking at all cases, you see that the only nontrivial elements of $A_6$ whose centralizers are nonabelian are those of the form $(ab)(cd)$, product of two disjoint $2$-cycles. If $x_{i,j}=(ab)(cd)$ then $x_{i,k}$ and $x_{k,j}$ must be elements of its centralizer and again of the same type. But the only such elements are $(ab)(cd)$, $(ac)(bd)$, $(ad)(bc)$, $(ab)(ef)$, and $(cd)(ef)$. There are only a few possibilities for pairs of these whose commutator is $(ab)(cd)$. You'll find that none of them will lead to a solution, a choice of $x_{i,j}$ for all $i$ and $j$.