I read that my Canon Speedlite 430EXII could only be used as a slave on my new 5D Mark 3 and that I should buy a new flash.
That's simply wrong. Any Canon EX-series speedlight (and many third party flashes) will work just fine with your 5Dmk3 provided you use them on camera, or off camera with the right cable. Your 430EXII is a great flash -- it's powerful with plenty of useful features, and moderately priced.
The thing that your 430EXII can't do is to control other Canon speedlights using Canon's master/slave protocol. Since you currently have just the one flash right now, that's not a problem.
Eventually, you'll want to move your flash off camera, and maybe even use more than one flash at a time. Canon's system makes that quite easy. There are cheaper alternatives, but if you decide to stick with Canon's system you'll want to pick up a flash that can work as a master. These are mostly the top of the line (or previously top of the line) flashes (600EX, 600EX-RT, 580EX, 580EXII, 550EX). However, it turns out that the diminutive (and cheap!) 90EX can also function as a master on bodies that are modern enough to have on-camera speedlight controls (including your 5Dmk3). So, for around $50, you can easily use your 430EXII off camera. That adds a lot of creative control (when you're ready for it) at a pretty reasonable price.
Obligatory link: If you're starting to get interested in flash photography, go read Strobist: Lighting 101. (Careful, you might get hooked.)