The polynomial $K[X]$ is of the form:
$$p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m}$$
where the $p$'s are elements of some field $K$ and $X$ are from elements of a ring $Y$.
I'm wondering about the operations between the field $K$ and the $Y$. Specifically we have multiplication $p_mX^m$. Since $K$ is a field, it has multiplication defined for elements in $K$. Likewise, $Y$ has multiplication defined for elements in its ring. But wondering where the multiplication is defined between $K$ and $Y$, since they are elements from different sets. Wondering if this means that you have a third multiplication defined for between $K$ and $Y$. So then you have $KK$, $YY$, and $KY$. I haven't read anywhere the rules for how this inter-set operation is supposed to be defined and if it differs from the single-set multiplication operation.
Also wondering if $p$ and $X$ can be from the same field.
There is no ring $Y$ involved here. Instead, $X$ is treated as a formal symbol, and you multiply expressions $$p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m}$$ formally using the ring axioms. In particular, for instance, the outcome when you mutliply an element $a\in K$ by $X$ is just $aX$: that's an expression of the form above with $m=1$, $p_0=0$, and $p_1=a$.
To be completely precise, an element of $K[X]$ can be defined as a function $p:\mathbb{N}\to K$ such that $p(n)=0$ for all but finitely many $n$. We think of this function as representing the expression $$p(0)+p(1)X+p(2)X^{2}+\cdots$$ (where the sum eventually ends because $p(n)=0$ for all but finitely many $n$). Addition of polynomials is defined by $(p+q)(n)=p(n)+q(n)$, and multiplication is defined by $(pq)(n)=\sum_{m=0}^np(m)q(n-m)$. This multiplication formula may look strange but it is exactly what you get by multiplying polynomials like you would in high school: to find the coefficient of $X^n$ in $pq$, you add up all ways to make an $X^n$ term by combining an $X^m$ term from $p$ with an $X^{n-m}$ term from $q$.
Thinking of polynomials as functions in this way, "$X$" is just a shorthand for the function $p$ such that $p(1)=1$ and $p(n)=0$ for $n\neq 1$. An element $a\in K$ can be considered as a "constant polynomial" by identifying it with the function $p(0)=a$, $p(n)=0$ for $n\neq 0$. Multiplication is then defined just like multiplication for any two polynomials, as above.