According to Rolling Stone magazine, http://www.rollingstone.com/music/lists/tom-petty-50-greatest-songs-w506653/learning-to-fly-w506739, quote:
"Petty claims to have gotten the words almost verbatim from watching a television interview with a pilot. "He said there's not much to learning to fly; the difficult thing is coming down, and I thought, 'Yeah, that's true,'" Petty recalled. The song, which also contains allusions to the just-ended Gulf War ("The sea may burn"), was pulled together in an evening with (Jeff) Lynne, who came up with chords to go under Petty's melody. It became one of his most inspirational tunes. "I wanted that song to be a kind of redemptive song," Petty said. "Only in the vaguest way."
So in the most basic sense, the narrator is learning to fly the way any human without wings has to learn, in an aeroplane.