One example is the Hurewicz theorem which tells us that (e.g) a CWcx with only one 0cell has a nontrivial fundamental group if H_1 is nontrivial. What other examples are there? (The CWcomplexes I have in mind always have exactly one 0cell, 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.)

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 DeligneGriffithsMorganSullivan, 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). 


If you don't want to make any assumptions about $\pi_1$, then I think the question is hard. Maybe Hurewicz is most of what you can say. If assume something like simply connected, you can say a lot more. There are many things you can say rationally, some of which were pointed bout my Kevin, but even integrally or mod p there are a lot of techniques. Can you be more precise about the setup you are interested in? You can always replace a connected CW complex with a homotopy equivalent one that has a single 0cell, so this is not really an issue. 

