The original WoW was painful. Mobs took forever to die; one additional enemy in a fight was a pain, two probably meant death. There was a ton of running. Most buffs lasted two minutes, many took reagents, abilities were trained and often out of reach if you lacked the necessary gold. Warlocks had to farm shards, hunters had to carry ammo -- even my warrior did, since in vanilla WoW she can carry a bow and fire arrows herself.
The quickest way to level up is to travel through the game's zones, completing all the quests you encounter along the way. Completing quests grants you Experience points, which will slowly fill in your experience bar with blue or purple. When the bar fills, you reach the next level. Alternatively, you could choose to run Dungeons or Battlegrounds, which are not covered in this beginner's guide.
A new quest will pop up once your first pet reaches level 3. It will send you to either Audrey Burnhep in Stormwind or Varzok in Orgrimmar. These trainers will offer a quest chain to battle specific Pet Tamer NPCs on each continent. Defeating each of these NPCs will reward a nice chunk of experience for your pets, as well a Sack of Pet Supplies (which has the chance to drop the Porcupette companion, among the always-useful pet bandages).
Things like Tauren melee reach being bigger, which is something I experienced going away live during Black Temple Illidan progression (at least for that particular encounter), is confusing to players as it's a significant advantage over other races in some instances. No quest tracking is another apparently confusing issue, as many Mankirk's WIfe seekers well know, and there's a whole lot more where that came from:
There are two parts of the quest chain for each continent. The first part involves defeating numerous Master Pet Tamers, and the second requires you to triumph over the Grand Master Tamer. After you've bested the Grand Master Tamer, dailies from each Tamer will unlock for that continent. All the dailies continue to reward experience for your companions, however only the Grand Master daily quest will reward the Sack of Pet Supplies.
I recreated my very first character -- a human warrior, because in the last-push alpha test I joined in 2004, there was no Horde -- and logged in. Immediately, I was surprised by how good the graphics actually looked, for being 15 year old textures-on-polygons. Warcraft's bright colors and cartoony aesthetic continue to this day, so all the increased resolution and better-contoured characters in Lordaeron don't really change the game's visual aesthetic.