First, while $\beta $ particles were identified early in the study of radioactive decay, the classic $\beta $ particle is a high-energy electron, part of $\beta $ - decay. Such an electron does not usually produce $\gamma $ radiation.
On the other hand, $\beta $ + decay produces a positron, and this will quickly encounter an electron, the two will annihilate, and a pair of 511 keV photons will be produced. And yes, 511 keV is $\gamma $ radiation.
As to your larger question, the answer is, "sort of, but not really". Generally speaking, $\gamma $ radiation is at the end of many decay chains, so you can sort of consider it the last in the sequence, except that electron/positron pair production occurs when a 1.022 MeV + photon hits an atom, so there's no absolute rule.
$\alpha $ (helium nuclei) and $\beta $ (electrons or positrons) processes are not ordinarily sequential, so it's not appropriate to talk about one occurring before the other.