There's no estimate that works in general. Krengel, "On the speed of convergence in the ergodic theorem", shows that for any ergodic transformation of $[0,1]$ and any sequence $(a_n)$ converging to $0$, no matter how slowly, there is a set $A$ such that $$\lim_{N\rightarrow\infty}\frac{1}{a_N}|A_N\chi_A(x)-\mu(A)|=\infty$$ for almost every $x$, where $A_N=\frac{1}{N}\sum_{i<N}f(T^ix)$. (The same function $\chi_A$ has similarly slow convergence in the $L^p$ norm for all $p\in[1,\infty)$ as well.)
As Vaughn Climenhaga has already pointed out, many additional assumptions have consequences for the rate of convergence. If you can't work with one of those assumptions, there's a big literature on the oscillations of the ergodic averages, both as "oscillations" and as "upcrossings", and those results suffice for many purposes.