I am in search of a function/equation, that gives me a graph similar to the one shown below.
I want $y$ to start increasing slowly at first, then ramping up faster and after a certain range, after Point $P$, it increases slower.
When $x$ goes to infinity, I want $y$ to approach $z$ but never really reaching it.
I would like to specify $z$ in the equation and also I want to know how I can manipulate in which rate $y$ increases/when the turning point (Point $P$) occurs, where the increase of $y$ slows down.
As a side note: I dont really care what happens for $x < 0$.
You might be looking for a logistic curve.
Perhaps try
$$f(x) = a\left(\frac{e^{ b \left(x-c\right)}}{e^{b \left(x-c\right)}+1}\right).$$
This will have the turning point $P$ at $x=c$, and will approach $a$ (but never reach it) as $x \to \infty$. $b$ can be used to control how steep the curve is. Working out $\frac{d}{dx}f$ at $x=c$ will give you a formula for how steep the curve is at $P$.
Further, if you want to control the $y$ intercept (which it seems you might) then try
$$f(x) = (a-y')\left(\frac{e^{ b \left(x-c\right)}}{e^{b \left(x-c\right)}+1}\right) + y',$$
which will have a $y$-intercept at $f(0) = (a-y')(\frac{e^{-bc}}{e^{-bc}+1}) + y'$.
From Desmos: