Understanding CLIQUE structure
Your understanding is right. So in particular, for any graph $G$ with a clique of size $k$, you will need at least $k$ colors to properly color it. In case this is not obvious, you can think about it via a contradiction. So assume $G$ contains a clique of size $k$, and you have a proper coloring with less than $k$ colors. It follows that the clique contains at least two vertices with the same color. By definition of a clique, these two vertices are adjacent. But this is a contradiction to the coloring being proper.

Further, one might also wonder if the chromatic number of a graph would always equal the clique number. In this direction, you can have a look at perfect graphs for which the chromatic number of every induced subgraph equals its clique number. On the other hand, the Mycielski graphs show that there are graph that don't even contain a triangle (equivalently, a clique on 3 vertices), while the chromatic number can be arbitrarily large.