I want a graph that goes from $[0, \infty) \Rightarrow (0, 1]$ a bit like $e^{-x}$, but less severe. By that I mean I do not want all the action around $0$.
The $y$ axis must start at $1$ and decay to, but not reach $0$.
I could obviously scale the $x$-axis, but it is still severe with most movement around $0$.
$f(x)=-\frac{2}{\pi}\arctan(ax)+1$
The rate of convergence can be controlled with parameter $a$. In general, this function will converge to $0$ much slower than $e^{-x}$, since the derivative of $\arctan$ is $\frac{1}{1+x^2}$. The $-\frac{2}{\pi}$ and $+1$ ensure that the graph meets your stipulated conditions.
Edit: Start by thinking of functions with horizontal asymptotes which converge more slowly then the exponential function. Then, adjust so they meet the parameters you're looking for. Another example would be $y=\frac{1}{x}$. Adjusting to meet your requirements: $f(x)=\frac{1}{ax+1}$. Choose $a\in(0,1)$ as desired.