iOS app “next” key won’t go to the next text field

The return key doesn’t do anything special in a text field by default. You need to explicitly change the first responder:

- (BOOL)textFieldShouldReturn:(UITextField *)theTextField {
    if (theTextField == self.textPassword) {
        [theTextField resignFirstResponder];
    } else if (theTextField == self.textUsername) {
        [self.textPassword becomeFirstResponder];
    }
    return YES;
}