The mod p cohomologies of classifying spaces of compact Lie groups I want to do some computation which need the mod p cohomologies of classifying spaces of connected compact Lie groups as input. I need the table for both the simply connected case and the central quotient case($G/\Gamma$, $\Gamma\subset Z(G)$). Are there a complete table for all these results. 
 A: In the book of Mimura-Toda "Topology of Lie groups" you can find a lot of the computations you are interested on.
A: Most of the time, H*(BG,p) is a polynomial algebra, generated in the degrees of the Weyl group.
The exceptional cases are known as torsion primes. If G is simply connected and simple, there are no such primes in type A or C, they are 2 in types B (except B2), D and G, 2 and 3 in types E6 and E7, and 2, 3 and 5 in E8. A precise definition is in Definition 2.43 of "Parity Sheaves", by Juteau, Mautner and Williamson.
To get an idea of the cohomology in one of the unusual cases, there is
$$H^*(BSO_3;\mathbb{F}_2)\cong \mathbb{F}_2[b,w] $$
where the generators b and w are in degrees 2 and 3. This is not actually hard to prove via a spectral sequence once you have the guts to believe that the cohomology ring is this big.
For a general reference for computing this cohomology ring, look at "Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups" by Kac.
