Desk research against primary sources, with the issuing authority linked where a page states an official fee, rule or deadline.
The attested birth certificate is usually the document that takes longest to arrange if it has not already been attested at home, since attestation has to happen in the issuing country before the paperwork is useful here — start that process before you start packing, not after you land. Report cards covering the last 2 years exist to place your child in the right year group, so a school may ask follow-up questions if those two years show an unusual gap or change of curriculum.
For British schools, the head teacher's reference letter is the document families most often forget to request, because it has to come directly from the outgoing school rather than being something a parent can supply. Request it at the same time as the transfer certificate rather than afterwards, since chasing two separate documents from the same office at two different points is slower than asking for both together.