If you mean a pet that can fight for you ... that's generally the domain of a Hunter, although you can collect non combat pets (generally small and piddly, like a parrot, or a fluffy, useless something-or-other). So, if you want the "big cat thing that you see running around next to people" as a warrior, then, sorry - no - roll a hunter for that.
In terms of Mounts, there's a rare Tiger mount from on of the level 60 raids that anyone can get (time and patience allowing). There's also a white tiger-type-thing you can get from a level 60 area (takes a lot of time and energy to get that as well, but you can do it on your own if you're a high enough level).
If you're an alliance character, then you can get your reputation up to the max level with Darnassus (I think - not being alliance, I don't know for certain), then you can get a tiger-thingy mount as a matter of course - this will put you on Par with every other alliance character (does alliance even have another type of mount ... anyone ?? ;->)
Edited, Jul 11th 2008 8:56am by robertlofthouse