They are wrong. You do not have to turn off cellular data to use wifi. Most phones have a wifi icon that appears when you are connected to wifi.
Most phones have a data use logger that shows which apps are using data. If there is a certain one that is using lots of data, through your settings, you should be able to set that app to use wifi only and not use cellular data. I know this can be done on the iPhone, I would think that most smartphones would have that feature.