I struggle to understand what the difference is between $\in$, $\subseteq$, $\subset$ and how they interact with different types of sets.
For example say I have two sets, $$A = \{ \text{red}, \text{green}, \text{blue}, \text{yellow} \}$$ and $$B = \{\text{blue}, \text{yellow}\}.$$
If I want to say "Yellow is inside of $B$" I think $\text{yellow} \in B$ is a correct way to do it.
But what if I want "$B$ is inside of $A$", is $B \in A$ still correct?
And what if I start using power sets? Are $B \in \mathcal{P}(A)$ and $\mathcal{P}(B) \in \mathcal{P}(A)$ correct, or should I use another operator?
$A\in B$ means that $A$ is an element of $B$. For example $2 \in \{1,2,3\}$, $4\in \mathbb N$, but we can also do that with sets, for example, $\{1\}\in\{\{1\},\{1,2\},\{1,2,3\}\}$. In this case, $B$ is always a set, and $A$ can be a number, a word, a set, etc...
when you write: $A\subseteq B$, both $A$ and $B$ are sets. This means that every element of $A$ is also an element of $B$. for example: $\{2\}\subseteq\{1,2,3\}$, but also $\{1,2\}\subseteq\{1,2,3\}$.
the diference between $\subseteq$ and $\subset$, is that, when we say $A\subseteq B$, this means that every element of $A$ is in $B$, or $A=B$ (it's the same when you write $x\leq y$. this means that $x$ is less than $y$, or $x=y$). When you write $A\subset B$, you are saying that every element of $A$ is in $B$, but $A$ is not equal to $B$ (just like when you right $x<y$).
Answering some of your specific questions: $B\in \mathcal P(A)$ means that $B$ is in the set, of all subsets of $A$, thus $B\subseteq A$.
$\mathcal P(B)\in \mathcal P(A)$ is a bit more tecnical. In general $\mathcal P(B)\in \mathcal P(A)$ is not correct, but $\mathcal P(B)\subseteq \mathcal P(A)$ is true if $B\subseteq A$.