I'm learning about indexed sets and came across the definition of an indexed set to be, "...any set,such that for $i \in I$, we have a set $A_i$ ...how would this definition be written using quantifiers? Please let me know if this definition is wrong as well...thank you!
I think it should look like this? but I'm not sure:
$$ \underset{i \in I}{\forall} \exists A_i $$
Your idea is right, even if your notation (setting the $i \in I$ underneath the $\forall$) looks a bit strange to my eyes.
If we want to codify the idea that "for each $i \in I$ there is a set $A_i$", then we would write
$$ \forall i \in I . \exists A_i $$
As an aside, this can be "implemented" in any number of ways.
Perhaps most naturally as function $A_{(-)} : I \to \mathsf{Set}$ which takes each element $i$ to the set $A_i$, but (for technical reasons) we often try to avoid functions into $\mathsf{Set}$ (since the set of all sets doesn't exist).
This leads to a related, if more difficult to motivate, implementation (called a "bundle") where we view the family as a single set $A$ with a map $p : A \to I$. Then we write $A_i = p^{-1}(i)$. This feels backwards, but turns out to be a very useful perspective to have on indexed sets!
I hope this helps ^_^