There are some simple such examples if you want to know the generators but not explicit equations for the surfaces:
- A general quartic in $\mathbb P^3$ has Picard number $1$ and hence its Picard group is generated by any minimal degree curve.
- A quartic, general among those containing a fixed line has Picard number $2$ and its Picard group is generated by that line and the hyperplane class (or the complementary cubic curve).
- A quartic, general among those containing a fixed conic has Picard number $2$ and its Picard group is generated by that conic and the hyperplane class (or the complementary conic).
- A quartic, general among those containing two fixed skew lines has Picard number $3$ and its Picard group is generated by those lines and the hyperplane class.
If you are only interested in generating $\mathrm{Pic}\otimes\mathbb Q$, then you have that
- $\mathrm{Pic}\otimes\mathbb Q$ is generated by the smooth rational and elliptic curves on it for any $K3$ with Picard number at least $5$.
- $\mathrm{Pic}\otimes\mathbb Q$ is generated by the smooth rational curves on it for any $K3$ with Picard number at least $12$.