Apple released a quick update, iOS 9.0.1, that seems to solve the major problem of iOS 9 — freezing your iPhone and iPad upon install. They also fixed a few minor bugs. It’s safe to update your devices. However, I have found some apps a bit flaky, sometimes becoming non-responsive. If you want to wait for a more stable update, I have read that 9.1 is being tested and will be available in a few weeks. If you do decide to update now, please make sure that you backup your device to your computer with iTunes or to iCloud before proceeding.
Bottom line: It’s safe to install but there may be some minor annoyances. It’s not perfect.