It definitely dose in my opinion at my age, I don't really know what replaced it unless I have a situation in where you would use it. In the U.S. we don't really go looking for things on the internet to pass time, we simply where on or looking at our social media like twitter or snapchat. In what situation would you need to use " surf the internet "?

It does sound a little like something a corny old teacher would say trying to relate.

Err... that's a really good question. "Surf" sounds like something cool and exciting (and suitable for when the internet was new), but I think "browse the internet" has become the more popular phrase. Still, "surf the internet" isn't embarrassingly outdated. At least to my mind.