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][1] in which we describe how to do this (again in much greater generality than you need).


  [1]: http://arxiv.org/abs/0809.4412