I'm on one of the first few quests where I'm supposed to make a bow, kill an animal, and cook the food. However, I can't find plant fiber anywhere to make the bow with. Where can I get some?
Some trees (which may not be present on every planet) yield plant fibre when cut down. Just use your matter manipulator.
Vines work as plant fibre. You can find those growing on cave ceilings. Cut them with the matter manipulator, too.
You can grow some yourself, which is renewable, but takes some time and requires some preparation. You'll need a few seeds (you start with those) and basic stone tools.