Allen Knutson said here in comments below the question that

I generally regard torsion in (co)homology as a sign that one should be computing K-theory instead, which has less of it.

I know one or two things about torsion groups, examples of cohomology groups that has non zero torsion part, some definitions $K$-theory.

Can some one help me to understand why torsion in cohomology should remind about computing $K$-theory? Any references are welcome.

Some not very well posed question is : what are (some of the) other signs that one should be computing K-theory?