Do I need an internet (data) connection if I want to use Fore on the golf course?

You must have an active data connection so that the courses nearby are displayed.

If you want to use the app without data connection, you must have downloaded the course data before (e.g. Wifi in clubhouse). All you have to do is start a round with the app and end it after the app has loaded the course data and shows you the distances immediately. 

You can do this, for example, in the clubhouse (advantage here, the place is also displayed directly), in the hotel (but the course may not be the nearest, so watch out which course you want to play) or already at home) because you have to search for the course by name. But please try different spellings, because sometimes the name of the course is a little bit different to the name used by its members (like a "course nick name").

You don't need a data connection while playing around and you can even switch your phone to flight mode. This has the further advantage that you a) have less battery consumption and b) will be not disturbed.

Course that you have already played or downloaded will be saved on your phone and you can find them under "stored courses".