If nobody is picking you up in surigao, I suggest you walk out of the airport and cross the street (this should take you about a minute or two). Tricycles are lined up. If you want to save, you can wait for other passengers and you pay about PHP10. But you can hire the whole tricycle for PHP50. The downtown area is about 3km from the airport.
The multicabs (small jeepneys) that are parked inside the airport will charge PHP200.
If you come from Manila, the logical way to Surigao is fly to Butuan and then catch a bus to Surigao since when there's bad weather in Surigao, plane would hardly land.
If you come from Cebu, if youre not in a hurry, take the boat. The Cokaliong boat leaves at 7 pm and arrives in Surigao 5 am. You can buy your ticket at their Cebu port office.