One example is the Hurewicz theorem which tells us that (e.g) a CW-cx with only one 0-cell has a nontrivial fundamental group if H\_1 is nontrivial. What other examples are there? (The CW-complexes I have in mind always have exactly one 0-cell, but for the sake of a more wide discussion one could assume this is not generally true. However, I am mostly interested in the higher homotopy groups as I already know everything about the fundamental group.)



What can be said about the homotopy groups of a CW-complex in terms of its (co)homology?
Try looking up some references on rational homotopy theory. Rational homotopy theory studies the homotopy groups tensor Q, so basically you kill all torsion information. If we focus only on homotopy groups tensor Q, the question you ask becomes easier. As Steven Sam mentions in the comments, the homotopy groups of spheres are really crazy. But the rational homotopy groups of spheres are quite tractable (in fact completely known, by a theorem of Serre) and can be more or less obtained from cohomology, if I recall correctly. 


One particularly impressive theorem, of Deligne-Griffiths-Morgan-Sullivan, says that if your space is a compact Kähler manifold (e.g. a smooth complex projective variety), and if you know its rational cohomology ring, then you can compute for instance the ranks of all of its homotopy groups (maybe you need an extra assumption that the space is simply connected or has nilpotent fundamental group). 


