surfing on web
surfing in web
surfing on the web........... makes sense
surfing in the web............. less common but also makes sense
or other variants? .............yes .........surfing the web
surfing the web
This is the idiomatic form. It means "To look at Web sites on a computer."
surf on the web
When you "surf" you ride on top of the waves on a surfboard. That is why "on" is the normal preposition for the activity of web surfing.
It is a particular kind of web, so the definite article "the" is used.
surfing in the web
When you do go surfing, usually you go surfing in the ocean. This variant is also possible if you have a good imagination. To me, it implies that you are deep into the worldwide web.