Abelian Group Element Orders I want to show that if a finite abelian group has elements of order $m$ and $n$ then it will have an element of order $\text{lcm}(m,n)$.

First I proved the lemma if $a$ has order $m$ and $b$ has order $n$ with $m,n$ coprime, then $ab$ has order $mn$. This is because the subgroups $\langle a \rangle$ and $\langle b \rangle$ have trivial intersection (because the order of their intersection must divide their orders $m$ and $n$, which is a consequence of it being a subgroup of both), which implies $a^{i} \in \langle b \rangle$ iff $i \equiv 0 \pmod n$, and similarly $b^{i} \in \langle a \rangle$ iff $i \equiv 0 \pmod m$. Using that we deduce that $(ab)^i = a^i b^i = 1$ iff $i \equiv 0 \pmod {mn}$.

So if $a$ was an element of order $m$ and $b$ and element of $n$ with $g = \gcd(m,n) \not = 1$ I thought that $\text{lcm}(m,n) = \frac{m}{g}n$ so and $\frac{m}{g},n$ are coprime so I should construct an element (from $a$) with order $\frac{m}{g}$ then conclude the theorem by the lemma. For the construction I think it's just $a^g$.

I just have a nagging doubt about the correctness of the second proof, did I miss some important detail? Also if there are any neater ways to prove this (which don't depend on the structure theorem) I would like to learn them too.
Saying that $m/g$ and $n$ are coprime is wrong. Consider $n=12$, $m=18$, the standard example. What is true is that $m/g$ and $n/g$ are coprime. But if you want to complete the proof using the elements $a^g,b^g$, then you will realise that although these now have coprime orders, the least common multiple (=product) of those orders is no longer $\def\lcm{\operatorname{lcm}}\lcm(n,m)$, but $\lcm(n,m)/g$.

So instead another idea is needed. The best I have been able to think of is treat all prime numbers $p$ dividing $nm$ separately; if $p$ divides the order of only one of $a,b$ then there is nothing to do for $p$, but if it divides both orders, then keep the element in which the multiplicity is highest (choose one in case of a tie) and kill the factors $p$ in the order of the other element (say it was $b$) by replacing $b$ by $b^{p^i}$ (where $i$ is the minority multiplicity). This modification is designed to leave the least common multiple of the orders of $a,b$ intact, and when all primes have been processed the greatest common divisor has become$~1$. After these modifications, the element $ab$ has the desired order.
Everything you said seems fine.  Using additive notation, you only need to consider the subgroup $sa + tb$ of $G$.  This group contains the element $a+b$ which has the property

$$ \hbox{lcm}(m,n) \cdot (a + b) = 0 $$

so the order of $a+b$ is at most $\hbox{lcm}(m,n)$.  

Moreover the order must be equal to $\hbox{lcm}(m,n)$, because any smaller order could be divisible by both $m$ and $n$.