I'm not sure how to use correctly $$\nabla \cdot \vec{F} = \frac{1}{r^2} \partial_r (r^2 F^r) + \frac{1}{r \sin \theta} \partial_\theta (\sin \theta F^\theta) + \frac{1}{r \sin \theta} \partial_\phi F^\phi$$
For exemple, I have $\vec{F}$ = $r^3\phi sin(\theta)(\hat{u_r} + \hat{u_\phi} + \hat{u_\theta})$
Do I just replace $F^r$ with $r^3$ and so on then find the derivative?
Is this just a plug and go formula?
Yes you simply take partial derivative as stated in the formula. In this case,
$\vec F = r^3 \phi \sin \theta \ (\hat{u_r} + \hat{u_\phi} + \hat{u_\theta}) = r^3 \phi \sin \theta \ \hat{u_r} + r^3 \phi \sin \theta \ \hat{u_\phi} + r^3 \phi \sin \theta \ \hat{u_\theta}$
Now simply take derivative for each component.
$\frac{1}{r^2}\partial_r (r^2 F^r) = \frac{1}{r^2}\partial_r (r^5 \phi \sin \theta) = 5r^2 \phi \sin\theta$
$\frac{1}{r \sin\theta}\partial_{\phi} (F^{\phi}) = \frac{1}{r \sin\theta}\partial_{\phi} (r^3 \phi \sin\theta) = ?$
$\frac{1}{r \sin\theta}\partial_{\theta} (\sin \theta F^{\theta}) = \frac{1}{r \sin\theta}\partial_{\theta} (r^3 \phi \sin^2 \theta) = ?$