Hi guys,
I work at CERN, not at the Visit service but perhaps I can help you with this. I can tell you that is not possible to visit any underground facility right now, if you don't have special permissions or "connections".
To visit CERN the best option is the free tour, it is most days at 10:30am. Best thing is to send an e-mail to the address Bertrand gave (visits.service@cern.ch) and ask if they have a tour on the date you want, and in your prefered language. If you are part of a big group a special visit can be arranged, even in other languages.
There are also 2 permanent exhibitions but they are mostly thought for children. You can take a look af them too if you come for the other visit, they are for free.
This regular visit consists in a 1/2hour video followed by a visit to an overground (does this word exist?) facility. The most typical is ATLAS (one of the most important experiments) control room, but it can change depending on the calendar and availability. The guides are normal workers who take a 1-day course, so don't expect too much from us please
By the way, don't expect too much about the research centre itself... It is just a normal work place, full of old and ugly buildings. It is not a place of wonders, innovative modern stuff, as some people think. I warn you because some people are disappointed because of that.
Feel free to ask!