I use the internet quite a bit, e.g. to send/receive email and to access the web. If I spend some time looking at web pages, then I will normally describe this as "looking at web pages" or perhaps "browsing the web". I do not use the phrase "surfing the net".
I recall a quote from Terry Pratchett: "You can't surf a net - you'd get all tangled up." That's true, but it's not my main concern. To explain my views on the subject, I need to digress into history.
A couple of hundred years ago, it was standard for upper class people (men and women) in England to wear a lot of white make-up on their faces. If you've seen "Blackadder III" then you'll get the idea. The reason behind this was that suntans were considered to be a bad thing - basically, if you were a common labourer, out digging ditches all day, then you'd get quite a tan. By contrast, if you were upper class then you could laze around inside all day, and stay quite pale. If you weren't naturally pale, then you'd try to disguise it, to avoid association with the οι πολλοι. More recently, that changed, and it became a sign of wealth that you could afford to go jetting off on foreign holidays to lie on the beach, so suntans became a good thing. (I suspect that the trend may be reversing again at the moment.)
There's a parallel between this and computer literacy, or at least there used to be. When I was at secondary school (1985-1992) it was actively considered a bad thing to know how to use a computer. I remember one of the other boys paying me to type up an English essay for him, just so that he could avoid the stigma of being seen in the computer room. This applied to technology in general - adults would cheerfully admit at parties that they had no idea of how to program their VCR. At best, technological expertise was seen as irrevelant. At worst, it singled you out as a target of ridicule. I remember a beer poster that I saw a while back - it was written in "leet speak", but it said something like "Not for computer nerds, for people with lives".
So, not a perfect situation, but at least it was simple - you had the division between people who enjoyed spending time on a computer and those who didn't. I've been using the internet since early in 1993, but back then it was quite a niche; very few people outside universities had even heard of it. Gradually this changed, and the nay-sayers realised that they actually wanted to use the internet after all. But this posed a dilemma - how could they do this, without tarring themselves with the same brush, and feeling like geeks?
Solution: new terminology! Surfing is pretty cool, and nobody's going to associate that with poor social skills. So, let's apply that phrase here - it's equal but different, like Bernard's irregular verbs in "Yes Minister":
I surf the net.
You are online.
He is a nerd who needs to get a life.
The closest I've ever come to surfing is watching "Point Break", but I've done a bit of snowboarding and I'm guessing that they're vaguely similar. More to the point, if I had to describe surfing, I'd say that it involves dedication and balance, and that it can be exhilerating as well as making you feel "at one with the wave". I really don't think that clicking a mouse to move to a different web page is anything like it, and I suspect that most real surfers would agree. In fact, if someone actually did sit in front of the computer shouting "Whoa!" and clinging onto their keyboard every time they went to a different website, I think they'd seem even worse than the stereotype they were trying to avoid.
So, I don't use that term. Spending time on the web or chatting on IRC may not be the best use I could make of my time, but I do at least call it what it is. It's not something I'm proud of, or ashamed of, it's just something that I do.