I have to show the following claims:
A simple graph $G$ with $|V(G)|\ge 6$ either has a $K_3$ or an independent set of size $3$.
A simple graph $G$ with $|V(G)|\ge 14$ either has a $K_4$ or an independent set of size $3$.
For the first one we consider vertex $x \in V(G)$. Then $x$ has $5$ options to have neighbors and by Pigeonhole Principle vertex $x$ has at least $3$ neighbors or at least $3$ nonneighbors. If any of its neighbors share an edge then we have a $K_3$, otherwise we get an independent set of size $3$. I think this is correct.
How should I begin to do the second one? Can it be done in a similar way?
Use the general strategy of the first statement, as well as the first statement itself, to prove the second statement. Also, you should be able to get away with $|V(G)| \geq 10$ in the second question.
Really you’re doing the first steps in inductively proving these kinds of things.