Leia is an accomplished pilot, as you'd expect of someone with latent Force potential and multiple years of training.
Someone’s prepping for a trip, she thought. Her mind worked at lightspeed, and she saw her escape plan unfold as clearly as if the Emperor himself had rolled out a carpet for her. Yes—a small thrill of victory raced through her. She could work with that. Her spirits lightened for the first time in hours, and she felt the crushing pressure lift off her chest. The shuttles would be fueled. And the shuttles were outfitted with weapons. She could blast her way out, and by the time they realized what was happening, she’d be through Tatooine’s atmosphere.
Take that! she wanted to shout to the other senators. She was about to prove exactly what she was capable of when given the opportunity to try. Her membership in the Rebellion still felt new, too fresh. She’d needed the mission to show them her dedication and how far she’d go for them if only they’d give her the support to get there. This—the story of her escape under Vader’s nose—would solidify her bond with them that much more. No one—not the media, not her aunts, not even her father—would be able to deny that she was a fighter and deserved to have her voice heard.
If I don’t get shot out of the sky first, she thought. No—she could do it. She’d had years of flight training. And, well, there were all those dunes to hide in. Let’s see how Vader liked getting gritty sand in sensitive places in his armor.
A New Hope: The Princess, the Scoundrel, and the Farm Boy
Flying a rustbucket like the Falcon would seem to present some element of difficulty (due to Han's under-the-counter modifications) but not to the point where she wouldn't be able to fly it.
You might want to note that the new (canon) junior novelisation for Empire Strikes Back suggests that Han simply gave her control in the escape scene and expected her to know how to fly. There was no "teaching" but she still did a good job of it.
Han jumps from the captain’s seat. “Take it!” he shouts at Leia.
He ignores her and runs to the mechanical port, where Chewbacca is already yanking at wires and crying in confusion. (Which sounds like “Arrrrraaaaragh!”)
“Oh, boy,” Han mutters. He slides into the pilot’s seat, relieving Leia of her duties—which she performed more than admirably, it should be said, for the TIE fighters are now some distance behind—and stares at the death trap that stretches out before them.
The Empire Strikes Back: So You Want to Be a Jedi?