I would say use Redux as soon as you realize that React's built-in state management is becoming a prop drilling hell.

Have a look a the React docs for the very useful technique of lifting state up. With this technique, you:

1. Identify the "common ancestor" component where the state should be maintained.
2. Pass methods down to lower-level components via props.
3. Have the lower-level components hook up those handlers to any events they fire (e.g., input text changing).
4. Update the state in the method.

This can get you very far in React. You'll know when it becomes a pain to manage.

Take a look at this post from Dan Abramov about when you may not need Redux.