I really like the book of Bott and Tu for the De Rham theory. Hatchers book - freely available on his site - contains nice treatments of singular and cellular (co) homologies.
About your comment. What relations between the theories are you looking for? The Eilenberg Steenrod axioms - http://www.encyclopediaofmath.org/index.php/Steenrod-Eilenberg_axioms - show that the singular, cellular, and de Rham theories are the same (you have to be a bit careful with the coefficients of course), on spaces where they are all defined. I believe the book of Bredon discusses this a bit, but I don't have it with me here (there is a very short passage in Hatcher). Group cohomology can be seen to be the cohomology of a certain space associated to the group. I does not really matter in which theory we compute these, because they will give the same results. I'm not familiar with the Daubault and sheaf cohomology, so I don't have anything relevant to add to this.