Method 1 (preferred)
On a computer, go to https://appleid.apple.com/ and select "Manage your Apple ID". Log in with your Apple ID and password, then select Password and Security from the left column.
Fill in the answers to your security questions and click Continue. You need to know the answers to your existing security questions before you can change them or create new questions.
If you do not remember your answers, there may be a link on that page where you can have a reset email send to your "Rescue email address", as long as you provided one in the past. This option will not appear if you have not already provided a Rescue email address.
About the Rescue email address: http://support.apple.com/kb/HT5312
Check all your known email addresses for their response, which will contain further instructions to follow. The email is automated and should arrive immediately. Check your spam / junk folder if you believe you did not receive it.
Method 2 (you cannot perform the above and you're stuck)
If there does not seem to be any way to reset your security questions, you need to contact Apple.
Navigate the columns as follows: All Products and Services > iTunes > iTunes Store > Account Management > iTunes Store account securityAnswer the questions and if required sign in with your Apple ID and password - same one you used to log in on this support forum.
You will find an option to send an email to Apple on that page. Compose a thoughtfully worded email explaining your dilemma, and politely ask to have your security questions reset. There are over 400 million iTunes account holders and Apple receives a great many of these questions every day, so make sure yours is easily understood.
You should receive a confirmation email immediately. Check your email spam / junk folder for it because a humanoid should respond in a business day or two, who will probably request you provide some more security-related information before your questions can be reset. You will have to wait for another reply which will provide instructions for resetting your password and your answers to the security questions.
Normal business hours apply, so if it is Friday you may have to wait until Monday.
These back-and-forth emails may result in a week of correspondence before it is fixed.