In my notes and online it seems to explain that for a multivariate function $z= f(x,y)$ you should know some functions $x(t), y(t)$ such that you can use the chain rule $ \frac{\delta{z}}{\delta{t}} = \frac{\delta{z}}{\delta{x}}(\frac{\delta{x}}{\delta{t}}) + \frac{\delta{z}}{\delta{y}}(\frac{\delta{y}}{\delta{t}})$ but I can't seem to understand how to get the function with respect to $t$ if I wasn't given it.
An example: $z=ycos(x)+y^2$ then $x(t)=t^2+s$ and $y(t)=2t^3$ ??
Example 2: $z=x^2y$ it was given that: $x(t)=cos(t)$ and $y(t)=sin(t)$ but what if I didn't know that?
I forgot a lot from this particular class and I have no idea how that works. What is the setup for finding $x(t), y(t)$?
You would need to be provided some additional information in order to say how $x$ and $y$ depend on $t$. You might be given explicit formulas for $x(t)$ and $y(t)$, or you could, for example, be told that $x(t)$ and $y(t)$ parameterize the unit circle in the $xy$-plane, starting at $(1, 0)$ and moving counterclockwise at a speed of 1 radian per second. In that case, you could deduce, using your knowledge of trigonometry, that $x(t) = \cos(t)$ and $y = \sin(t)$.
However, if you aren't provided any information about how $x$ and $y$ depend on other parameters, then the most you can do is express $\frac{\partial z}{\partial t}$ in terms of $x$ and $y$. For instance, in your example $z = x^2 y$, you could say that $$ \frac{\partial z}{\partial t} = 2xy \frac{\partial x}{\partial t} + x^2 \frac{\partial y}{\partial t}. $$