Warriors in human history books tend to be leadership figures, such as Anduin Lothar, Danath Trollbane, and even the current king of Stormwind, Varian Wrynn (son of Llane Wrynn, also a warrior). If your warrior character follows this tradition, he or she is likely to try and be a leader lots of the time, leading other people into battle and generally acting very brave, even against terrible odds. You may call yourself by the title of "Knight" or "Champion," (especially if you have one of those titles left over from the old PvP system).
Then again, those common everyday guards in Stormwind are warriors too, so your character needn't necessarily be very ambitious. You might even take some inspiration from the quests various guards will give you in Elwynn, Duskwood, and Redridge settings, and just decide to be a random soldier trying to get by in the world, practicing with your trusty sword on the target dummies, sweating under the weight of his armor, getting sent to all sorts of strange places on various assignments just like people do with soldiers in the real world. The more you can make use of military jargon, the better.
Dwarves are generally a more focused lot than humans, being much longer lived and usually more patient too. A dwarf is likely to have spent hours upon hours training and sparring with axe and hammer, and are far too proud to complain about the weight of their armor -- being genuinely related to the Earthen stone people of ages past, they probably don't even feel the weight that much anyways. The armor probably feels like an extension of their own skin in ways they can't really describe. Your dwarven warrior might see himself as a great explorer who makes the way safe for his people's archeologists, or as a noble defender of his people's homeland and the principles it stands for.
King Magni Bronzebeard is a warrior, along with his brother Muradin Bronzebeard, who was thought killed by Arthas in Northrend. Kurdran Wildhammer might be a good example of a more feral sort of dwarven warrior, since he used to be Thane of the Wildhammer clan, who are widely known for their daring acts of bravery... and foolhardiness too -- one dwarf's glorious death against impossible odds might seem like an idiotic charge into a wasteful death in the eyes of a human, but oh well. That's what you get with some cultural differences in Azeroth. Incedentally, a dwarf warrior is also the race and class most likely to try fighting while drunk.
Night elves are unique in that their warriors are more likely to be female than male, at least according to tradition. These female warriors continue to be known as Sentinels, especially if their main task is to defend the forests of Ashenvale from intrusion. Nowadays the Sentinels are part of the Alliance, however, and they may not be so accustomed to working with strange peoples in far away places when the situation requires it. A night elf warrior prizes speed, agility, and quickness with their weapon more than brute strength or foolhardy charging into danger. Cunning, planning, and precise execution are the ways of a night elf warrior, not vainglorious shows of false valor.
Strangely, even though Sentinels were primarily female, it seems a lot of the more leader-like warrior figures in Warcraft lore were male, such as Kur'talos Ravencrest and Jarod Shadowsong (though these examples were mainly from the time before so many night elf men were called away to other duties in the Emerald Dream, which opened the door for more women to become warriors). Even Illidan Stormrage was a warrior before he became the first demon hunter. [Edit: oops. He was a caster, not a warrior as such, but in becoming a demon hunter, he became very much like a warrior, I would say.] Your character, too, might hope to become a demon hunter someday, especially if he thinks Illidan was kind of cool. Night elf women warriors have a few great role-models, however, such as Shandris Feathermoon, the general of all Sentinels, stationed at Feathermoon Stronghold in Feralas, and of course the woman who tracked down Illidan to the very gate of death: Warden Maiev Shadowsong.
Of all races and classes in World of Warcraft, gnomish warriors are the feistiest. When you're that small, you can't tolerate any fear in the face of enemies so much larger than yourself. Still, gnomes had to fight a lot in their ill-fated defense of Gnomeregan, so it makes sense that there would be quite a few gnome warriors who no longer fear the big creatures of the world -- or have been driven so mad by the terrifying realities of war intruding on their otherwise gentle demeanor that they just don't give a poopie about death anymore.
Gnomish warriors take great advantage in being so small, without ever admitting that they are small at all (the other races are just very large, you see). They easily dodge blows and weave their ways under their enemy's legs to cripple them from below, striking at their feet, their legs, or even that most tender and vulnerable of places on the body.
Gelbin Mekkatorque, leader of the gnomes, could be viewed as a warrior, though his first love is for tinkering. Like him, it would make sense for a gnomish warrior to have a completely different persona he wears in times of peace, only to surprise everyone with his ferociousness once push comes to shove.
Of all the races who train to be warriors, draenei are the least likely to brag about their prowess with weapons, to look forward to any sort of battle, or to even think of themselves as "warriors" in the same sense that other races do. The draenei whose job it is to defend the Exodar see themselves as "Peacekeepers" rather than fighters in grand battles.
One of the reasons that there are so few notable warriors in draenei lore is that draenei seem to view combat abilities as an unfortunate necessity rather than a praiseworthy achievement. For them, true glory is attained through more peaceful pursuits, such as learning, prayer, and harmonious society. They fully recognize, however, that the peace must be kept, sometimes through violent means, and they prepare themselves for such violence without shame or hesitation.
Next week, we'll have a look at the races of the Horde and their quite different views on warriors, battle, as well as a few of their most famous warrior heroes which your character might view as a role model.