The Philippines is served by two main international hubs: Ninoy Aquino International Airport (NAIA) in Manila and Mactan-Cebu International Airport in Cebu. Both receive direct flights from much of Asia, the Middle East, and the US West Coast. From Europe, you will nearly always connect in the Gulf states or Singapore.
Budget carriers fly in from across Southeast Asia — Cebu Pacific and AirAsia cover routes from Bangkok, Kuala Lumpur, Singapore, Hong Kong, and Tokyo, often cheaply. Philippine Airlines and Cathay Pacific handle the longer-haul routes. Clark International Airport (north of Manila) handles some budget regional routes and is worth checking if you're heading straight to the Cordillera.
Most nationalities enter visa-free for 30 days. You'll need a passport valid for at least six months beyond your stay and an onward or return ticket — immigration officers do check. Extensions are easy to arrange through the Bureau of Immigration, with 29-day increments available up to 59 days, and longer-stay visitor visas after that. Citizens of a small number of countries require a visa in advance; check the Department of Foreign Affairs website.
All travelers must complete an eTravel registration (free, web-based) within 72 hours before arrival. There is no longer an arrival card to fill in on the plane.
Arriving overland from another country is not really possible — the Philippines has no land borders. A few inter-island ferries connect to Malaysia (Zamboanga to Sandakan), but this route has been disrupted historically and is worth checking carefully before attempting.