Public transport from both the Manila and Cebu international airports requires changing jeepneys several times and isn't advisable given that taxis are so cheap.
Ninoy Aquino International Airport (NAIA; Manila) Yellow metered taxis are the best option for getting into town. They are cheap (P225 average to most hotels) and usually plentiful at all four NAIA terminals. Uber is another option, with fares averaging P150 to P200 (off-peak). Check carefully which terminal you arrive at (and, especially, depart from) – you'll need up to an hour to switch terminals.
Mactan-Cebu International Airport (CEB; Cebu) You’ll find a taxi rank of regular metered taxis (P40 flag fall) on the right as you exit. Uber is also an option. Either costs about P300 to the city centre.