jmc

Rather than doing long division, we will write the given fraction to have a denominator of the form $10^b=2^b\cdot 5^b$, where $b$ is a positive integer. First, we write $\frac{3}{1250}$ as $\frac{3}{2^1\cdot 5^4}$. To make the denominator fit the form $2^b\cdot 5^b$, we make $b$ the larger of the two exponents, which in this case is $4$. Thus, we have $$\frac{3}{5^4\cdot 2^1}\cdot \frac{2^3}{2^3}=\frac{3\cdot 2^3}{5^4\cdot 2^4}=\frac{24}{10^4}$$The exponent in the denominator is $4$, and $24$ are the last two digits. Therefore, there are $4-2=\boxed{2}$ zeros between the decimal point and the first non-zero digit.