Making sure I understand the power of Lagrange's Theorem Does Lagrange's Theorem lead to the result that if |G| = 20, for example, then I can expect to find subgroups of orders 1, 2, 4, 5, 10 only or just that those are some expected subgroup orders?

If $|G| = 20 $, it means that if $H$ is a subgroup of $G$ then it is necessary that $|H| \mid |G|$. However, note that Lagrange's theorem doesn't at all guarantee the existence of subgroups with orders that are divisors of 20. That is, Lagrange's theorem doesn't guarantee the existence of a single non-trivial subgroup. Therefore Lagrange's theorem doesn't assure us of the existence of subgroups with orders 2,4,5 and 10. But it does guarantee that if we happen to find any subgroup of $G$, then it will necessarily be a divisor of 20. Remember to not confuse Lagrange's theorem with its possible converse, which is false.
