$$F=G \frac{m_{1}n_{2}}{r^{2}}$$
Note that $r$ represents the exact distance between the centers of the masses. For example, if you have to consider the gravitational attraction between the sun and the moon, you have to consider:
Sometimes we can assume things to be point masses.
Gravitational field - region in which you experience a force
More concentrated field lines $\implies$ greater field strength
If $F=G \frac{m_{1}m_{2}}{r^{2}}$ then $g=\frac{GM}{r}$
Where $g$ is the gravitational field strength in $N kg^{-1}$
Note that $g$ for Earth is at Earth's surface, but will get weaker as you go further above the Earth's surface.
$g=\frac{GM}{r}=\frac{6.67\times10^{-11} \times 5.97\times 10^{24}}{(6371\times10^{3})^{2}}=9.81036Nkg^{-1}$.
Using these formulas should be enough to approach most questions, including ones with unique (made-up) planets, and the classic question of an object in between two bodies.
