Most Apps need access storage (to write something) or internet. Usually you should not concern about this. If the app cannot access internet, no access to storage, it can do nothing.
If an app requires access your contact, location information, or messages, then you need check whether the app is reliable (e.g., read the reviews).
Android is based on Linux, so the privilege control is very effective and well-tested. Normally, if you don’t root your phone, the app will not able to get full control of your phone. Of course, some malicious apps may try to read the pictures and files, but usually they cannot access private info (e.g., contact list, which is manged by Android system).
The bottom line is to install apps from reliable sources (e.g., Google Play store) only. Unless it is necessary, do not root your phone.
Android is more secure than Windows phone, or iOS.