Solution:

The numbers, arranged from smallest to largest, are $\log_{3} 2$, $\frac{2}{3}$, $\log_{625} 75$, and $\log_{5} 3$.

- Since $\left(3^{\log_{3} 2}\right)^{3} = 8$ and $\left(3^{\frac{2}{3}}\right)^{3} = 9$, then $\log_{3} 2 < \frac{2}{3}$.

- Since $\left(625^{\frac{2}{3}}\right)^{3} = 5^{8} = 5^{6} \cdot 25$ and $\left(625^{\log_{625} 75}\right)^{3} = 75^{3} = 5^{6} \cdot 27$, then $\frac{2}{3} < \log_{625} 75$.

- If $A = \log_{625} 75$, then $5^{4A} = 75$. On the other hand, $5^{4 \log_{5} 3} = 81$. Thus, $\log_{625} 75 < \log_{5} 3$.