Are there any regulations the FAA has ruled to get into a Control Tower? Do people need any special kind of security clearance? Are there forbidden items or other special rules?
I'd also want to know if there's a way for non-pilots (your average civilian Joe) to get into a Control Tower.
It is still possible (and in my experience not that difficult) to get FAA tower tours (and sometimes even TRACON or Center tours), though you should be reasonable in your expectations here -- You are probably NOT going to get a tour of the tower at JFK or LAX.
The FAA used to actively encourage pilots with an instrument rating (or pursuing instrument training) to sign up for ATC facility tours under a program called "Operation Raincheck" -- This program is still running, though it's been substantially scaled back (Poke around on the FAA Safety Team's SPANS system, or sign up for notices in your area and keep a sharp eye out. Tours usually have a very limited number of slots available, and they fill up fast!).
(As an example of how quickly the slots fill up, when N90 (New York TRACON) last had a Raincheck event it was full within a few hours of being announced. No, I didn't get a slot.)
At towered GA airports - particularly ones with a lot of flight training - you can also contact the tower directly. The printed AOPA Airport Directory has facility numbers in it but that's a little out of date. You can also get the phone number from Flight Service, the local FSDO, or ask the tower/ground controller for the number if they're not too busy.
When you call the facility explain that you're a pilot/student pilot and would like to arrange to tour the facility (and if possible shadow a controller). They will usually ask for your name, address, airman certificate number, and a few other things to run a check and get approval to let you in to the facility. You will also be expected to have government issued photo ID with you when you show up.
My experience arranging a tour of our local tower was extremely pleasant - they are happy to do the tours and let you shadow a controller, and I even got to pick their brains about some peculiarities in local procedures.
It's a great educational experience, and I would recommend it for any pilot/student pilot.
As @voretaq7 says, getting tours of LAX of JFK towers just won't happen unless you happen to fly for an airline with a base there and/or are good at persuasion. I've tried to negotiate a tower tour with the supervisor at LAX on short notice but he did indicate if I had a lot of lead time they might have been able to work something out.
The easiest way to get a tour in my experience is just to call and ask for someone to talk to to request a tour. I've been in the tower cab at EWR, AUS and CRG and in the Austin TRACON. All of those were arranged with a phone call, a social security number and proof of citizenship (all post 9/11). It helps to mention you are a pilot and want to see how things work on their side.
Typically the tower guys are going to want at least one of you to be a pilot and they'll want more lead time if you are bringing a group. They'll also probably want to verify citizenship for everyone ahead of time and you'll want to be up front if you have an non-US people in the group just so they don't get surprised when you show up and they ask for passports or other ID. Bring donuts.
I have also toured the San Angelo, TX FSS. For those guys I was already on field and just called the briefing number and asked if I could walk over. They were happy to have us (I had a student with me).
If you want a tour of any facility, just call. They'll tell you what they require and how and when the tour will happen and who you can bring.
I've arranged tours of two control towers (a class D and a class B) and a TRACON without any issues. I was only required to provide proof of US citizenship, from what I remember.
You must turn OFF (not airplane mode) cell phones before entering the tower or active part of the TRACON.
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