Children are usually the most adaptable members of an expat family, and also the ones whose struggles hide longest, so both truths deserve attention. The broad pattern: younger children absorb new languages and norms with almost unfair speed, often out-speaking their parents within a year, while teenagers face the harder road, since the move costs them established identity and friendships at exactly the age those matter most, and they merit extra patience and consultation in family decisions. Most children of all ages find their feet within a school year; a silent adjustment period, some regression or clinginess early on, is normal rather than alarming.
Schools carry most of the integration load, and good ones have machinery for it: newcomer language immersion classes in many public systems, buddy programs pairing new students with established ones, and international schools whose entire population understands arrival and departure. Ask any prospective school specifically how they support new foreign students, since the quality of that answer tells you a lot. Outside school, the reliable engines are sports teams, music and arts classes, scouts and youth groups, and neighborhood play, all of which build friendship through repetition; community centers and libraries often run affordable options.
For parents, two additions: your own calm modeling of adaptation matters more than any program, since children read your stress accurately, and keeping the home language and culture alive is a gift, not an obstacle, because bilingual, bicultural kids ultimately count their upbringing as an advantage.