With Wrath of the Lich King Classic Pre-Patch now live, the way to use mounts has changed in the old Classic TBC realm, with the addition of the Mount Journal.
With the pre-patch, mounts and pets now have their own collection journals. You can use your mounts directly from your Mount Journal after you learn the mount. With this change, mounts and pets no longer use bag slots as they did in Classic and The Burning Crusade.