You have been told (or heard) slightly incorrectly.
Some apps use internet data and some don't.
What kind of phone do you have, out of curiosity?
If your app wants data there are basically two ways it can get it depending on how your settings have been set.
It can get the data by wifi or by what is called "Cellular Data" by Apple. It is the cellular data, the data from cell phone (mobile) masts which will send you to the poorhouse. Wifi data will cost you only what you have to pay the hotel or coffee bar for the wifi - mostly in Europe now that is free.
With an iPhone the main settings screen offers you as the first option next to the orange icon with the while plane on it the chance to enter "Airplane Mode". That turns off all the phones radios and will prevent you receiving or making any calls, and using any cellular data or any wifi.
If you turn on airplane mode you can still tap the next button down and turn on wifi and log in.
Be careful because if you reset your phone or turn it all the way off it will switch everything back on when you restart the phone.
You should use airplane mode when on a plane to comply with the law in most countries.
You can also switch off airplane mode to restore the phone to normal and tap the 4th button down in the settings "Cellular" and then switch off Cellular Data. That will leave everything else, including phone calls, in the normal mode.
So... to cut to the chase, it is apps which use Cellular Data which you need to be careful about, wifi is no problem, and instead of disabling apps you can cut out cellular use or cellular data use depending on your needs and the plan you have.
The chance is that the person you spoke to has never traveled out of the country and is trying to get it right but just missing the target.
And best news of all - the Rick Steves App uses no data at all once you have downloaded the walks you want, which you have already done. It just works like an mp3 player and plays the files already on the phone.
If you have a different operating system - such as Android, Blackberry, or Windows Phone, etc., the settings will be slightly different, and the iPhone Rick Steves app is miles ahead of the Android one. The iPhone one has just been upgraded, those changes have not yet happened on the Android app.
That's why I asked you to share the type of phone you have, and if you can what release of the operating system is on it.
I'm sure you'll be fine..... we'll give you everything you need to know.