This turned out to be a trickier question that I imagined. After an hour of Googling, I was able to find out what are the Internet usage stats per country, how fast do servers respond in various parts of the world, and even (though this data is pretty old) how much data the entire Internet contains.
The closest I got to the answer was at the sites of the big analysts - Nielsen/Netratings and comScore, which has the approximation of world internet usage of people over 15 years of age (why the age restricion, I don't know - I guess they're marketing oriented so they're interested in buying power more than anything else).
This number, cited from comScore's site, which I reckon represents the number of people who use the internet regularly, is 712,976,000. Yes, that's a big number, but it's not exactly what I was looking for. Besides, www.internetworldstats.com says the number is 1,086,250,903. On the other hand, Nielsen/Netratings seems to be more conservative, citing their "Current Digital Media Universe Estimate" at 482,821,770.
What I (mostly) don't know is the methodology behind this data - do these people use the internet daily, once every week, or, for example, once every month? Things get more complex if you want to count the unique hits the internet gets, because one person can connect, disconnect and reconnect several times per day, and every time they do that it would technically be a unique visit. To complicate matters further, it would be interesting to know the sum of daily unique hits on all websites in existence.
However, with the help of this little chart, I can at least approximate these numbers.
Bear in mind that I used data from different sources, obtained with different methodologies and from different samples, and I did some huge simplifications to make my life easier. The numbers in this final paragraph are very rough and possibly very wrong.
So, if the average person has approximately 33 internet sessions per month, that would mean approx. 1.1 every day, which would mean that the internet gets somewhere between 531,103,947, 784,273,600 and 1,194,875,993 visits every day, depending on whose numbers you trust.
According to Nielsen/Netratings, an average person visits 1446 websites per month. (1446/30) = 48.2*482,821,770= 23,272,009,314 (if you trust Nielsen/Netratings) or 48.2* 1,086,250,903 = 52,357,293,525. (if you trust internetworldstats.com). This is the number of webpages visited daily.
Looking at these numbers I can't help the feeling that I might have screwed them up for a billion or two. Looking forward to constructive criticism (;.