Manin's book "Cubic forms" contains  the calculations of these groups when $X$ is a smooth projective cubic surface.  In particular, $Pic^0(X)=\{0\}$ and $Pic(X)=NS(X)$ is a free commutative group of rank 7.

Another class of  examples is provided by products $X=E \times E'$ of two elliptic curves. In particular, if $E=E'$ has no complex multiplication then $NS(E\times E)$ is a free commutative group of rank 3 generated by the classes of $E \times \{0\}$, $\{0\}\times E$ and the diagonal while $Pic^0(E\times E)=E \times E$). See Mumford's Abelian varieties.