Short answer
You can fly to the Azores from mainland Portugal (Lisbon and Porto) in about two hours, or from other European cities and North America on direct flights, mainly to Ponta Delgada (São Miguel) and Terceira. There are also inter-island flights. Ferry services connect some islands but are limited and seasonal. Always check schedules and book in advance, especially in summer.
General information, not legal or tax advice. Rules and amounts change: confirm with official sources.