Taming Quests
The most important quest your character will complete is the taming quest. When done you will be able to tame beasts in the wild and raise them to be your ally.
Each race will get their own version of the quest. To start you will need to talk to a specific hunter trainer in your level ten starting area. He will give you a taming rod and tell you to use it on a creature nearby. After you use the rod the monster will start attacking you until the taming is complete. Make sure you are at full health and have aspect of the monkey up. If anyone steals the mobs attention by damaging or taunting it the taming will fail. After you have tamed the first target he will send you out after two more. When you have brought the final one back the hunter will give you your Tame Beast ability and a follow up quest. The follow up quest will have you visit a hunter trainer inside your capital city. You can find them quickly by simply asking a city guard where the hunter trainers are. After you talk to them they will give you your Feed, Revive, and Train Pet abilities. Now you can go out and tame your first real pet.
Dwarf
Begin by heading to Kharnos and talking to Grif Wildheart. He can be found standing on a small hill between the inn and the blacksmith shop. His first target is a Large Craig Boar which you can find directly east of Kharnos. Bring it back to him to get your next target, a Snow Leopard. You can find them directly south of Kharnos. Your final target is an Ice Claw Bear, which you can find either directly east or directly south. After you have brought the last pet back Grif will give you your Tame Beast ability. Now you need a quick run up to Ironforge to learn how to take care of your pet. Head to the Hall of Arms and look for Belia Thundergranite among all the other hunter trainers. She will teach you how to feed, train and revive your new pet.
Night Elf

Begin by finding Dazalar in Dolanaar right behind the Inn. His first target is a Webwood Lurker which can be found all around Dolanaar, especially in the south.

Your next target is a Nightsaber Stalker. They can be found south of Lake Al'amaath. Your final target is a Strigid Screecher. Travel west to the gates of Darnasus and look for them on either side of the road. After taking it back to Dazalar you will learn your Tame Beast ability. Now accept the follow up quest from Dazalar and head to Darnasus. Look for felmale night elf named Jocaste among the hunter trainers in the Cenarion Enclave to learn your feed, train and revive pet skills.
Draenei
Start by talking to Huntress Kella Nightbow on Azuremyst Isle. She is outside the Exodar, near the dock off of the island. Her first target is a Barbed Crawler, which can be found along the coastline, especially south near Silvermist Isle. Your next target is a Greater Timberstrider. You can find lots of them north and west of the Exodar. The final target is a Nightstalker which you can find in the same area as the Timberstriders. For the follow up quest head into the Exodar and run to the south western most part of the Trader's tier and talk to Ganaar. He will teach you your feed, train and revive pet spells and give you a new crossbow.
Orc/Troll
Start by finding Thotar in Razor Hill. He can be found inside one of the burros across from the giant Barracks. His first target is a Dire Mottle Boar, which can be found just outside the town. His next target is a Surf Crawler. Head south to Senijin Village and swim over to the echo Isles until you find one on the beaches. The final target is an Armored Scorpid. You can find them by heading directly west along the road until you reach an open area. Bring it back to receive your follow up quest. Travel to the Valle of Honor and look for Ormak Grimshot among the hunter trainers. He will teach you your feed, train and revive pet abilities.
Tauren

Begin by talking to Yaw Sharpmane in Bloodhoof Village. He can be found inside one of the small tents. His first target is an Adult Planestrider. You can find lots of them to the south east. Your next target is a Prairie Stalker. They can be found north across the lake and near the wells. Your final target is a Swoop. These buzzards can be found to the north and east. For the follow up quest head

to Thunder Bluff and talk to Holt Thunderhorn. He can be found inside the tent on the Hunter's Rise. Talk to him to learn your feed, train and revive pet abilities.
Blood Elf
Start by talking to Lieutenant Dawnrunner at the Farstrider Retreat. Her first target is a Crazed Dragonhawk, which can be found directly south of Farstrider. Her second target is an Elder Springpaw. You can find these near the retreat to the north east. Your final target is a Mistbat, which can be found on the northern most part of the Ghostlands. Bring it back to Dawnstrider to receive the follow up. Now head to Silvermoon and talk to Halthenis in the northern building of Farstrider Square. You will receive a new bow and your feed, revive and train pet.
The Green Drake
This is your level 52 class quest. At the end are some nice blue quality rewards that you will use till level sixty. First head to one of your hunter trainers and accept the quest The Hunter's Charm. Next head to Azhara in Kalimdor to find Ogtinx, a furlbog hunter. He is on the cliff side at 42, 42, directly north and east of the Ruins of Eldarath.
His first task for you is to collect a two perfect courser antlers. You can find loads of Mosshoof Coursers directly north on the top half of the isthmus. You will need two pieces to make one pair.
His next chore has you gather six wavethrasher scales. You can get these from wavethrashers and young wave thrasher hydra in the nearby ocean. You can find Wavethrashers directly south of Ogtinx and young wavethrashers north on the edge of the zone. Return back to him when you have six.
Your final task is the hardest. You need to travel to the Sunken Temple in the Swamp of Sorrow and kill the dragon, Morphaz, inside. You will need a group to reach him and clear half of the instance before reaching him. He is in the second level in the giant circular room, past the six trolls, and usually inside one of the tunnels. If you bring a level seventy player along you should be able to take him down. If not, you will need a full group to deal with him and his add.

After killing Morphaz head back to Ogtinx and turn in his tooth. For your reward you can choose a Hunting Spear with crit and mana per five. The Devilsaur Eye which will temporarily increase your attack power and

chance to hit. And the Devilsaur Tooth, which makes your pets next attack a guarantied critical strike. The spear is very nice for its level and will last you till you leave for outland. You can get better weapons on the AH if you have the cash. But you can find better weapons in Outlands in just a few levels so you might want to save your gold. The Devilsaur Eye is also very nice, especially since there are so few trinkets at this level. It is nice for high DPS fights, so if you intend to do instances before going to Outlands this might be a better choice over the spear. You will be able to replace it with two trinkets in outlands by level sixty two however. The Devilsaur Tooth is probably the weakest of the rewards. On extra critical strike by your pet is not very useful at any level.