  The weak field limit for a pointlike source of a $f(R) \propto
R^{3/2}$-gravity model is studied. We aim to show the viability of such a model
as a valid alternative to GR + dark matter at Galactic and local scales.
Without considering dark matter, within the weak field approximation, we find
general exact solutions for gravity with standard matter, and apply them to
some astrophysical scales, recovering the consistency of the same
$f(R)$-gravity model with cosmological results.}{In particular, we show that it
is possible to obtain flat rotation curves for galaxies, [and consistency with]
Solar System tests, as in the so-called "Chameleon Approach". In fact, the
peripheral velocity $ v_\infty $ is shown to be expressed as $ v_\infty =
\lambda \sqrt{M}$, so that the Tully-Fisher relation is recovered. The results
point out the possibility of achieving alternative theories of gravity in which
exotic ingredients like dark matter and dark energy are not necessary, while
their coarse-grained astrophysical and cosmological effects can be related to a
geometric origin.
