The index of intersection satisfies certain properties which are easier to apply than the two definitions you give. For instance, if the tangent cones at P (initial forms, if $P=(0,0)$) of $F_1$, $F_2$ have no common factors, the index of intersection is just the product of multiplicities at $P$. And the index of intersection of $F_1=0$, $F_2=0$ coincides with the index of intersection of $F_1=0$ and $F_2+G\cdot F_1=0$ for every $G$. You can find a list of the relevant properties, with examples on how to apply them, in the relevant section of Fulton's book on Algebraic Curves.
There are two other ways to compute this number. One is, as you say, resolve the singularities of the union $F_1\cdot F_2=0$; then the intersection index is the sum of the products of the multiplicities of (the strict transforms of) $F_i=0$ at all blown up points (this is due to Max Noether). The other is to parameterize all branches of one of the curves $F_1=0$ and substitute the parameterizations in the other equation $F_2=0$. If the parameterizations are minimal, the intersection index is the sum of the resulting orders. Both these ways are explained in Casas-Alvero's book on Singularities of Plane Curves.
The index of intersection satisfies certain properties which are easier to apply than the two definitions you give. For instance, if the tangent cones at P (initial forms, if $P=(0,0)$) of $F_1$, $F_2$ have no common factors, the index of intersection is just the product of multiplicities at $P$. And the index of intersection of $F_1=0$, $F_2=0$ coincides with the index of intersection of $F_1=0$ and $F_2+G\cdot F_1=0$ for every $G$. You can find a list of the relevant properties, with examples on how to apply them, in the relevant section of Fulton's book on Algebraic Curves.