Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of differential geometry, differential topology and algebraic topology?
Matthias Kreck and collaborators have set out to create an encyclopaedic catalog of manifolds and their properties a while ago (http://www.map.mpim-bonn.mpg.de/Main_Page), although I don't think it's quite at a stage where one might describe it as useful. Its aim is to collect all known information about "real-world" manifolds, whatever that means. It's a wiki, so if you're interested in helping assemble such a catalog, you might consider contributing. Personally I think that the world of manifolds is too rich and, well, manifold, to be usefully catalogued in this way, but I'd love to be corrected.
Such tables indeed exist even though only partial information is available, of course. Higher homotopy groups of $RP^n$ are the same as those of spheres, so look at tables here. For Lie groups there is a good survey of Mimura ["Homotopy theory of Lie groups". Handbook of algebraic topology, 1995] with some tables if memory serves. There are also tables for exceptional Lie groups. If you want a more basic info there is a monograph "Topology of Lie groups I, II" by Mimura and Toda.
In general, I recommend a mathscinet search on "homotopy" and "Lie group" in the title. A number of things has happened since the above-mentioned works, and other information is in paper (not table) form. See e.g. here.