Let $G$ be a group such that $G = \langle a, b \rangle$.
Mainly I wanted to see if $o(G) = \text{lcm}\left(o(a), o(b)\right)$, but I know $D_6$ is a counterexample. But I can't think of any counterexample to see if $o(G) \leq o(a)o(b)$. Is it true? If it is, how could I prove it?
As pointed out by others in the comments, $S_n = \langle \underbrace{(1 \dots n)}_{a}, \underbrace{(1 \, 2)}_b\rangle$. $n! = o(S_n) > o(a) o(b) = 2n$ for $n > 3$, and the dihedral groups work since they are generated by two distinct reflections.
For a non-symmetric or dihedral group example, consider $A_4$ with the presentation $A_4 = \langle \underbrace{(1 \, 2 \, 3)}_a, \underbrace{(1 \, 2)(3 \, 4)}_b \rangle$. $12 = o(A_4) > o(a)o(b) = 6$.