I know what homology means and I know what (de Rham) cohomology means. But I can’t actually calculate cohomology of a given space (while I can do it for homology)
My question is that is there a relationship between homology and cohomology? (I mean can I derive something about cohomology is I know its homology?) If not, how can I calculate cohomology?
de Rham cohomology is only defined for smooth manifolds. It it is known that it agrees with singular cohomology on smooth manifolds.
Now you can apply the universal coefficient theorem for cohomology. See for example https://en.wikipedia.org/wiki/Universal_coefficient_theorem or any textbook on algebraic topology. This theorem establishes a short exact sequence $$0 \to \text{Ext}(H_{n−1}(X);\mathbb{Z})\to H^n(X) \to \text{Hom}(H_n(X);\mathbb{Z}) \to 0$$ In many cases you can compute $H^n(X)$ using this sequence. However, in general it is an extension problem with a non-unique solution.