Since the order of $S_3$ is $6$, you can do the following: let $X$ be a subset. If $X$ contains an element of order $3$ and an element of order $2$, then it contains the subgroup they generate and so, by Lagrange's theorem, it generates $S_3$.
If $X$ does not contain an element of order $3$, then it contains only transpositions. If it contains only one transposition, then clearly $X$ generates a cyclic group, and $S_3$ is not cyclic. So it contains at least two. Now note that $(12)(23) = (123)$ (and similarly the product of any two distinct transpositions has order $3$), so again $X$ generates $S_3$.
If $X$ does not contain an element of order $2$, then it contains only element of order $3$. But there are two of those, and they are multiples of each other, so in this case $X$ can only generate $C_3$.
In the end, $X$ generates $S_3$ if and only if $X$ contains at least one transposition and a distinct nontrivial element.
In general, you can say that transpositions generate a symmetric group, or that a $n$-cycle and a transposition generate $S_n$, but the aim is more oriented towards finding a generating set, instead of all.
There are, however, probabilistic results that for particular classes of groups $G$ tell you what is the probability that any two, three, $n$ random elements generate $G$.