You can't go wrong writing unit and feature tests. Especially if you work with any external API - tests would help ensure it's working properly (allowing you to debug app-specific issues vs API).
I think that a lot of the issues you'd face with React Native development are hardware based, so E2E testing would be more important than unit testing. Though currently, I'm pretty sure there isn't a testing library that can mock/assert against the Android and iOS operating systems (like Chromium or Selenium for web browsers). Meaning automated or programmatic E2E testing isn't possible on mobile, requiring manual testing anyway.