After choosing a race, class and customizing the look of your character, you will spawn in your race's starting zone, usually within a small settlement. Typically, these beginning areas are designed to get you to around level 10 before you leave and move on to the next zone. Right away, you will notice that some NPCs have a yellow exclamation mark () over their head. These symbols will also display on the mini map in the top-right corner of the screen if you are close to them. Talking with these NPCs will allow you to read and accept a quest they have for you. The objectives and status of this quest can be checked at any time by opening the quest log (L). Once you have completed the objectives for said quest, you can return to the NPC (or possibly a different one) to turn in the quest. The NPC whom you need to turn a quest into will have a yellow question mark () over their head. As you complete quests and your character gains levels, always be on the lookout for new quests being offered by NPCs.
Keep in mind that these quests are optional, and are not necessary to continue leveling and capturing companions. They are also account-wide, so if you complete the quest chain on one character, you will not be able to complete it again on another. However, any progress that you've made will be saved and you can pick up where you left off in the quest chain on another character. The dailies that you've unlocked will also be accessible by all your toons.
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.
If a companion dies during the battle, it will need to be resurrected before it can participate in another encounter or be summoned as a traveling companion. Get that pet a medic! Or in this case, a Stable Master. Stable Master NPCs can heal and revive your pets for a small fee, and there is no cooldown on the Stable Master's heal. You’ll find Stable Masters in major towns and cities across Azeroth.
Where to get materials and reagents you may ask? Buy for cheap off farmers in bulk or track Auction House lots. Get in touch with most big sellers. Communication is important here, send a whisper or write them a letter. It would be good for both of you to establish business relationships. Constant suppliers and buyers is convenient for both parties to save the time and effort.