Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d?

e.g. for $d=4$ the cohomology lattice is $E_8^{\oplus2}\oplus H^{\oplus3}$, but what is the class of a hyperplane section in this lattice? Probably many things are known for $d=5,6$?
(I need the results for any smooth $S_d$, not necessarily generic.)

And what is known for "not-too-singular" $S_d$? (e.g. with nodes only)