First add your permissions to "android/app/src/main/AndroidManifest.xml" file. You can use npm module in order to achieve your goal.
npm install --save react-native-permissions
After installation, you can use checkMultiple function to ask multiple permissions. Following code illustrates how to ask permission for ANDROID.CAMERA and ANDROID.READ_EXTERNAL_STORAGE :
 checkMultiple([
  PERMISSIONS.ANDROID.CAMERA,
  PERMISSIONS.ANDROID.READ_EXTERNAL_STORAGE,
]).then(result => {
  
  console.log(result);
  
});
The result will be an object like this
{"android.permission.CAMERA": "denied", "android.permission.READ_EXTERNAL_STORAGE": "denied"}
To handle this result, you can use this basic code example
checkPermissions = () => {
if (Platform.OS !== 'android') {
  return;
}
checkMultiple([
  PERMISSIONS.ANDROID.CAMERA,
  PERMISSIONS.ANDROID.READ_EXTERNAL_STORAGE,
]).then(result => {
  if (
    !result ||
    !result[PERMISSIONS.ANDROID.CAMERA] ||
    !result[PERMISSIONS.ANDROID.READ_EXTERNAL_STORAGE]
  ) {
    console.log('could not get any result. Please try later.');
  }
  if (
    result[PERMISSIONS.ANDROID.CAMERA] === RESULTS.GRANTED &&
    result[PERMISSIONS.ANDROID.READ_EXTERNAL_STORAGE] === RESULTS.GRANTED
  ) {
    console.log('granted for both permissions');
    // do smthing here
  }
});
};