What is the best way to become a computer support specialist ?

That's a really broad term. Can you define it better?

A+ (http://www.comptia.org/certifications/listed/a.aspx) and Net+ (http://www.comptia.org/certifications/listed/network.aspx) certs then add a ACMT if you work with Macs (http://support.apple.com/kb/ht3431?locale=en_US). Then you may want to look into a Linux Cert (http://shop.canonical.com/index.php?cPath=21).
The certification is proof of a level of knowledge or training and though some of the best computer people I ever worked with were either not certified or under certified having your basic certs is a prerequisite to getting in the door in most cases. Most basic certs can be obtained through self training the more advanced certs often require instructor led classes

I think certifications are way over-rated these days. Although as cmeeks says they are unfortuantly a pre-equistite to getting in the door most places. The route I would recommend though would be to just donating time and more time in front of the screen messing around with system. Go to the question sections here, and if you find a question you don't know the answer to go research and learn till you do. That's what I did for years, and taught myself everything for free. Then after that go get some basic A+ style certs. To get you in the door.

I certaintly wouldn't bother wasting money on paying companies fortunes to studying the topics as mostly you can learn it for free faster and easier.

Some good places to start you off are:

W3Schools (http://www.w3schools.com/) (online programming tutorials HTML, PHP, Javascript, MySQL, ASP etc etc)
RhysHaden (http://www.rhyshaden.com/) (Great networking resources, can teach you almost anything you need)
TechTutorials (http://www.techtutorials.net) (There is some rubbish on it, but quite a few gems)
HellboundHackers (http://www.hellboundhackers.org/) (Don't let the name put you off. Great site for learning about security)