Question: How do I show that picking 3 socks containing just black and red socks that I must get either a pair of black or red socks?

I mean it's fairly obvious, but how would I show it?

Is this pigeon hole?

Answer: 
You could show it by brute force, listing all $8$ of the possible outcomes of picking $3$ socks: BBB, BBR, BRB, RBB, BRR, RBR, RRB, and RRR. However, that’s a little messy, and the pigeonhole principle provides a neater answer. You have two pigeonholes, one labelled BLACK and the other labelled RED. You pull out $3$ socks and put each of them into the appropriate pigeonhole. Since there were more socks than pigeonholes, one pigeonhole must contain more than one sock.
We can use the Pigeonhole Principle (strong form). That is $q\_1+q\_2+⋯+q\_n−n+1$ where $q\_i$ is the number of desired socks of a certain color, $(2)$ and $n$ is the number of colors, $(2)$. So $2+2-2+1=3$.