What is the difference between a Set, a Vector, and, a Vector Space?
A set is what's called a primitive notion. That means we don't actually define it, we just assume that everyone has roughly the same idea in mind when we say it. Think what a set of objects means outside of math. Same definition should work here. A set is a group(/ collection/ assortment/ assemblage/ ... gaggle -- maybe that one only works for geese) of objects. Those objects are called members or elements of the set.
A vector is a member of a vector space.
A vector space is a set of objects which can be multiplied by regular numbers and added together via some rules called the vector space axioms.
An example of a set is $\{1,2,3\}$. This is just a collection of three numbers. Another example is $\{banana, orange, taco, oatmeal\}$. This set contains $4$ foods. Neither one of these sets comes pre-equipped with any notion of addition or multiplication -- they're just groups of objects.
An example of a vector space is the set of all real numbers, $\Bbb R$, along with the usual kind of multiplication and addition. This is a set of numbers, but we also can talk about adding them and multiplying them by other numbers. You can check that they satisfy all of those rules that adding and multiplying have to satisfy and thus this set is also a vector space.
The number $3$ is an example of a vector in the above vector space. So is $\pi$. So is $\dots$
A less trivial (and more typical) example of a vector space is the set of all continuous functions defined on the interval $[0,1] \subset \Bbb R$. Addition of two functions $f$ and $g$ is defined by $(f+g)(x) = f(x)+g(x)$ and multiplication is defined by $(af)(x) = a\cdot f(x)$, for $a\in \Bbb R$. You can verify that this is also a vector space.
The function $f$ given by $f(x)=x^2-2$ is a vector is this vector space.