Learning how to determine the centralizers of semisimple elements of finite groups of Lie type?

Question

Suppose that $s$ is a semisimple element of a finite classical group of Lie type $G=G(q)$ such that $\lambda_1, ..., \lambda_n$ are eigenvalues of $s$ (counting multiplicities). My question is that how can we determine the structure and order of subgroup $C_G(s)$ (specially in cases that $G$ is of adjoint type or simply connected type)?

Someone referred me to the paper of R. W. Carter : Centralizers of semisimple elements in finite classical groups. As I skimmed the paper, the method mentioned there needs to know the theory of linear algebraic groups and so on. Since I'm quiet a beginner at algebraic groups, I want to know that whether there is another simpler approach to determine $C_G(s)$?

If not, I would be grateful If someone could guide me which sections of linear algebraic groups do I need to learn to be able to compute the centralizers? I have rather good background in group theory, some very basic definitions of linear algebraic groups, and almost nothing in theory of Lie algebras! So a detailed learning roadmap is highly appreciated. Thanks.