The key here is layers. It's fine to wear lightweight fabrics on a plane. It may even be a smart strategy if you're flying to or from a sweltering climate. But planes are often very cold—and blankets aren't exactly freely distributed on many flights these days. So fight the air-conditioned chill by layering up.
Getting warm? Remove a few layers, bundle them, and then use them as a pillow.
Bonus: The more layers you can pile on your body, the less clothes you need to pack in your suitcase.