Moving house — within 14 days
Take your residence card to the city/ward office of your new address within 14 days (転入届 when moving to another municipality, 転居届 within the same one). The office writes the new address on the back of your card — this also counts as notifying immigration. Moving to another municipality? File 転出届 at the old office first.
New employer or school — within 14 days
Mid- to long-term residents must notify immigration within 14 days when their organization changes:
- Work statuses (技術・人文知識・国際業務, 技能, 介護, 研究 …): 所属（契約）機関に関する届出
- Students and others (留学, 教育, 経営・管理 …): 所属（活動）機関に関する届出
You can notify online, by post or at the counter. Missing it can count against you at your next renewal.
Travelling abroad
With a valid passport and residence card you can leave and return within 1 year (or before your stay expires, if sooner) on a special re-entry permit (みなし再入国許可): tick the box on the embarkation card at the airport. For longer trips you need a re-entry permit (¥4,000 single / ¥7,000 multiple).
Sources: ISA — change of address, ISA — re-entry.