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Is there one short reference, which does a good job in providing a dictionary between algebraic group over a field and its root systems/datum?

I have a basic background with algebraic geometry and algebra. I am familiar with Weyl groups, tori, Levi and Cartan decomposition,... Nevertheless, I have a really hard time grasping the main observations involved in the “equivalence” of the categories.

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    No I have not, but I will have a look. Thx.2011-12-19

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You might find what you want in Kleshchev 's Lectures on Algebraic Groups, a free online course.
It is 160 pages long, but you can probably skip the first 90 pages which are a general introduction to algebraic geometry.
Root systems are explained from page 131 on.