As a website programmer, you are not required by law to have any qualifications, but some companies do ask applicants to have some sort of qualifications such as computer science or design and programming. Many programmers are self-taught or learn on the job as this is often the best way to pick up programming techniques and skills. Many employers look more at the experience a programmer has got rather than qualifications when it comes to offering job opportunities.
You can chose to do a computer programming course at college or university which will teach you a good overview of many different programming languages. If you continue on to become a web developer then you may find that you only ever program in one particular language anyway so this is not always necessary.
You can start off as a programmer by editing code that has already been written. Often then you can quickly advance to writing your own code and building websites from scratch. Some of the more advanced programmers even write their own content management systems.