Family hitting r-sets I'll start with a definition.
We say a family $\mathcal{F}\subseteq [n]^{(k)}$ hits every $r$-set for some $r\geq k$ if for each $R\in[n]^{(r)}$, there exists $F\in \mathcal{F}$ such that $F\subseteq R$.
First of all, I am tasked with finding the smallest family of 2-sets which hits all 3-sets in $[7]^{(3)}$.  I've found such a family of size 9, but how do I know it is the smallest?  If it is not the smallest, how do I find the smallest?  I assume it has to do with Turan's theorem, but I'm not sure I understand that well enough.
The method I used was using a greedy algorithm, but I don't think that always gives you the smallest such family.
 A: You are right about the connection to Turán's theorem.  It might help to consider the problem in a complementary form.
Suppose you have a family of $2$-sets that hits every $r$-set, and consider the complement of your family - that is, the collection of $2$-sets (which we may think of as a graph) that are not in your family.
Given any $r$-set $R$, we know that your family hits it, so there is some $2$-set in your family that is contained in $R$.  That means that $R^{(2)}$ is not contained within the complementary graph.  In other words, the complementary graph does not contain $K_r$.
Since you want the smallest possible hitting family, the complement should be as large as possible.  Hence the complement should be the largest $K_r$-free graph, which is the $(r-1)$-partite Turán graph.
For your particular parameters, the largest $K_3$-free graph on $7$ vertices is $K_{3,4}$, which has $12$ edges.  Hence the smallest hitting family will have size $\binom{7}{2} - 12 = 21 - 12 = 9$, matching your result (it should be a vertex disjoint union of $K_3$ and $K_4$).  Turán's, or indeed Mantel's, Theorem shows that you cannot do better.
