The procedure depends on whether the family already lives in Berlin, the child's age and what is already arranged. Start with the general plan so you do not repeat finished actions.
Answer the questions and get your own task list →If a family member enters later, first check the entry procedure by their citizenship, document and basis of residence. The general Berlin address registration does not replace a check of the entry rules.
Official source: German Federal Foreign Office: visas and family entry ↗A place in a funded kindergarten needs a Kita-Gutschein (voucher). The application can be filled in online, then the signed document is sent to the district Jugendamt (youth office) by post or handed in in person. Finding a place and issuing the voucher are different actions.
Official source: Berlin: the Kita-Gutschein service ↗The school search and enrolment depend on the child's age, address and previous schooling. The responsible district is determined by the address, and the procedure and form of registration are stated on the city's official page.
Official source: Berlin Welcome Center: schools ↗Check the insurance of each child and adult separately. Free family insurance in the statutory system is possible only if the conditions are met; the status is confirmed by the fund.
Official source: Federal health portal: insurance ↗NOSTRES is an independent service. The procedure and applicability depend on your situation; the final decision is made by the responsible authority or insurance fund.