This is well-known, and there are a number of relevant references. Firstly, there are these by Wall (they are pretty hard to read though).
Wall, G. E. Conjugacy classes in projective and special linear groups. Bull. Austral. Math. Soc. 22 (1980), no. 3, 339–364.
Wall, G. E. On the conjugacy classes in the unitary, symplectic and orthogonal groups. J. Austral. Math. Soc. 3 1963 1–62.
You might also be interested in this paper which I personally find much more readable.
Macdonald, I. G. Numbers of conjugacy classes in some finite classical groups. Bull. Austral. Math. Soc. 23 (1981), no. 1, 23–48.
You could also look at Carter's Finite groups of Lie type (email me if you want a copy) although that is a rather hard text and is much more general than you need.
For conjugacy in $PSL_3(q)$ you could also just do a search on Jordan rational forms - these forms classify conjugacy in $GL_3(q)$, and this classification can then be adapted to deal with $PSL_3(q)$. I have a paper with Anupam Singh in which we describe how to do this (again in much greater generality than you need).
