According to Carole Sampeck of The Lombard Archive, it was aviation pioneer and Hollywood mogul Howard Hughes, with whom she had once had a brief affair (http://community.livejournal.com/carole_and_co/11206.html), who got her interested in flying.
In the wake of Charles Lindbergh's successful trans-Atlantic solo flight in 1927, aviation was evolving from a daredevil's occupation into a fashionable mass pastime, and Hughes -- who apparently remained an acquaintance of Carole's after they broke up -- suggested she try it. Lombard, always interested in pursuing something new, took up his offer.
But, according to Sampeck, it never went beyond that. "It is my understanding that she did take it up as a lark, as several other actresses were doing at the time," Sampeck said. (Among actresses proficient in flying was Ruth Chatterton.) "She took it on with her usual verve and focus, of course...and when she got bored, she moved on." Sampeck doesn't believe Lombard ever owned a plane, for example.
Here's another pic of Carole near a plane:
It should also be noted that Carole took a number of airplane trips as a passenger, a few of them with Clark Gable, before that fateful flight from Indianapolis. Here are Gable and Lombard with fabled Hollywood pilot Paul Mantz, a close friend of Clark's:
In October 2006, Lisa Burks, a member of the popular Yahoo! group Hollywood Underground, visited Forest Lawn Glendale cemetery to place flowers near Lombard's crypt, as requested by a Hollywood Underground member who lives some distance from Forest Lawn. (She did.)
At the time, Forest Lawn was running a museum exhibit, "Icons of L.A. -- 100 Years of Forest Lawn and Los Angeles: History in Photography," Guess which pic was used to represent Lombard? That's right, the one directly above, showing her leaning against a plane.
As Burks wrote, "Hundreds of portraits of Carole exist, why this one? Is Forest Lawn completely clueless? Talk about inappropriate."
The entire entry can be found at http://lisaburks.typepad.com/home/2006/10/last_saturday_w.html