We calculate reduced $B(E2)$ electromagnetic transition strengths for light nuclei of mass numbers $B=8,12,16,20,24$ and $32$ within the Skyrme model. We find that the predicted transition strengths are of the correct order of magnitude and the computed intrinsic quadrupole moments match the experimentally observed effective nuclear shapes. For the Hoyle state we predict a large $B(E2)\uparrow$ value of $0.0521e2b2$. For Oxygen-16, we can obtain a quantitative understanding of the ground state rotational band and the rotational excitations of the second spin-0 state, $02+$.