When a group acts on a \(C^*\)-algebra \(A\), this induces actions on the Cuntz semigroup \(W(A)\) of \(A\) and on the \(K\)-theory group \(K_0(A)\). The present article studies to what extent important properties of the dynamics may be captured by these induced actions, which are much simpler than the original action. The article studies such notions of minimality, topological transitivity, and topological freeness. The author defines notions of \(K_0\)-minimality and \(W\)-minimality and shows that the action is \(W\)-minimal if and only if \(A\) has no ideals that are invariant under the group action. Any \(W\)-minimal action is also \(K_0\)-minimal, and the converse holds if \(A\) has cancellation and any ideal in \(A\) admits a non-zero projection. This also leads to a characterisation of minimality in terms of states on the ordered \(K_0\)-group of \(A\) if \(A\) is stably finite and all ideals in \(A\) contain non-zero projections. A group action on \(A\) should be called topologically transitive if any two non-zero invariant ideals in \(A\) have a non-zero intersection. In the presence of the intersection property, this is equivalent to the crossed product being a prime \(C^*\)-algebra. The intersection property says that any non-zero ideal in the crossed product has a non-zero intersection with \(A\). This article introduces variants of topological transitivity on the Cuntz semigroup and \(K_0\). The Cuntz semigroup version of this notion is related to the primeness of the crossed product. This suggests that it could very often be equivalent to topological transitivity as defined above. Finally, shadows of topological freeness of a group action on the Cuntz semigroup \(W(A)\) and \(K_0(A)\) are introduced. If \(A\) is commutative, then the Cuntz semigroup version of topological freeness is equivalent to ordinary topological freeness. If every non-zero hereditary subalgebra of \(A\) contains a projection and cancellation holds, then the \(K_0(A)\)- and \(W(A)\)-versions of topological freeness are equivalent.