Generators of the Symmetric group $S_3$

As James noted in his comment, generating sets are not unique, since if $A$ is a set that generates the group, then any set containing $A$ will also be a generating set. However, I assume you are trying to find a smallest set of generators.

If you allow an element $g$ to be a generator, then everything in the cyclic group $\langle g \rangle = \{1,g,g^2,\ldots \}$ will be taken care of. In particular, if you have one $3$-cycle, then you get the other one ($(1\,2\,3)^2=(1\,3\,2)$ and $(1\,3\,2)^2=(1\,2\,3)$). So if you have one $3$-cycle as a generator, you only need to get the three transpositions. By inspection, if you take any one of them as a generator, you can get the other two transpositions by multiplying it with the $3$-cycles.

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In general, for the symmetric group $S_n$, the following are generating sets. $$\{(1\,2), (2\, 3), \ldots, (n-1,n)\}$$ $$\{(1\,2), (1\,3), \ldots, (1\,n)\}$$ This also implies that the transpositions generate $S_n$.

You can generate $S_3$ with a rotation $(1\:2\:3)$ and a flip $(1\:2)$, think geometrically.

$S_3$ can be generated by a 2 cycle and a 3 cycle. For example $(12)$ and $(1 2 3)$.