A lot of the Horde and Alliance specific quests out here have the same rewards that are well worth pursuing. You'll be tasked to gather scrap metal, kill the local fel orcs, and burn the relic siege weapons of the opposite faction. Do all of these quests. An example would be the Horde quest Bonechewer Blood and its Alliance counterpart Fel Orc Scavengers, both of which give a nice shield which will probably be an upgrade if you leveled to 58 and came to Outland as soon as possible.
My first piece of serious advice to the warrior starting out in Outland is to avoid the Auction House. You won't need to buy any gear. There are several nice starting blues at level 60 which will tempt you: avoid them. You don't need them. Any slot that needs an upgrade will most likely find a good quest reward in this zone. There are plate helms, nice chestplates, starter greaves and shoulders that will last you almost all the way to 70, in my opinion. There are even green weapons that rival or eclipse particular old world epics. So save all your money and just gear up through quests and instances. The main instance in Hellfire is, of course, Hellfire Citadel, with two instances that will be of interest to you at this stage of the game. You'll start off running Hellfire Ramparts at around level 59 (at least I have seen level 59's run it) and will move up to Blood Furnace by the time you're nearly done with the zone. Both instances will have several drops that interest you. If you do the quests at Thrallmar/Honor Hold and run these instances a few times each, you'll probably be level 62 easily, especially if you move on to do the satellite quests at Falcon Watch/Temple of Telhamat.
A note: pick up Hellreaver if you can out of the chest at the end of Ramparts. Sure, you'll stumble across better, but look at it. It's pretty. Also, if you hit honored with your faction capitol while you're still in your low 60's, which you really should if you do all the quests available, you'll be able to buy either the Footman's Longsword or the Grunt's Waraxe, both useful for tanking with decent stamina and hit rating.
From here, you'll have a choice between two zones: you can head to Zangarmarsh before you're even done with Hellfire if you're bored (the quests there start around level 60, so it's even possible to skip Hellfire entirely if you're of a mind to, although I wouldn't recommend it) or if you're a completist and find yourself running out of quests around level 62, you can instead go to Terokkar Forest. Both zones will have plenty of quests including a rep grind of sorts (Zanga providing you with your first exposure to the Cenarion Expedition while Terokkar allows you access to Shattrath City and the competing Aldor and Scryer factions) and both zones also have winged dungeons to explore, with Zangarmarsh hosting the Coilfang Reservoir (where you'll be running Slave Pens and Underbog) while Terokkar hosts Auchindoun, which at this level has first Mana Tombs and then Auchenai Crypts. If you fully explore both of these zones and run the four dungeons you can easily be level 65 if not 66 by the time you even think about moving on from them, and whichever one you choose to explore first is more a matter of your level when you start than anything else. Zangarmarsh quests start a little lower than Terokkar, so my advice (based on what I did myself) is to plow through Zanga first, going to Terokkar as soon as you hit 62 if you're not already to get this sweet hat.
After these two zones you'll almost certainly be 65 if not 66 if you fully explore them, but if for some reason you get bored around 64 and want to move on there's still hope: you can go to Nagrand. There are faction specific quests here that give many of the same rewards, as Alliance find themselves aiding the Kurenai while Horde discover the mysteries of the Mag'har brown orcs. Horde warriors, it is imperative that you do every quest you can get your hands on in Nagrand. Alliance will find no such awesome quest chain waiting for them, certainly not one with such awesome rewards - I used those legs as PvP legs until I got the Gladiators and I was incredibly happy with the stamina, crit and +hit on them, especially when combined with a Clefthide Leg Armor. The high stamina and defense also makes them passable tanking legs. Alliance can take heart that there are still excellent rewards for them from the Nesignwary and Ring of Blood quests in this zone. Other quests with good rewards include the Durn the Hungerer slaying quest and the questline that ends with Gurok the Usurper. Once again my advice is to avoid the AH and simply clean this zone out. You'll still be running the Coilfang and Auchindoun instances at this point, and by the end of Nagrand you should be more than ready for the first of the Caverns of Time instances, Old Hillsbrad. There are many excellent drops for a 66 and up warrior in there, including the standout Durotan's Battle Harness. Get it as soon as you can. You'll also want to start building rep with the Keepers of Time as soon as you can especially if you intend to tank, as they have two excellent rewards for hitting revered.
You can easily be 67 by the end of Nagrand and from there, move on to Blade's Edge. If you play your cards carefully, try and stay rested and run as many instances as you can, it is even possible to reach level 70 without setting foot in either Netherstorm or Shadowmoon Valley, allowing you to use their rich lode of high level quests to fund your flying mount purchase. I wish I'd been this smart on my night elf, but both my human and tauren warriors managed to hit Netherstorm at 69 and to save the majority of those zones until after reaching 70, thus having more quests that paid well. With the addition of daily quests, this can become even more lucrative for you, combined with your having saved your gold by avoiding the AH as much as possible. While in Blade's Edge, both Alliance and Horde can ultimately end up killing a Gronn, but the Horde version is cooler because Rexxar is in it. You'll want to do it either way to pick up the Blackened Chestplate if Durotan's hasn't dropped for you. You'll make 13g for this one even if you don't want the chest, even more if you're somehow 70 when you do it. There are a lot of quests in this zone, including ones for the Cenarion Expedition which you want to build rep with (trust me on this) and which ends in a quest chain for a nice DPS or tanking neck.
By this point you should be ready for Sethekk Halls. Not only will you need to run this instance to get your later Shadow Labyrinth key, there's a quest that has some decent gold and XP and a blue polearm if you're really looking for one, although it's more of a hunter weapon. But quest drops are always free of loot drama. The standout drops in this instance are the Bands of Syth, Terokk's Nightmace, Deathforge Girdle and the Crow Wing Reaper, and they'll all last you well into being 70.
Also, if you have any aptitude for PvP at all, consider doing the daily BG quest. If you get it completed it's decent quest XP, almost 500 honor and you'll end up with marks and honor to spend at 70 to get immediate gear upgrades. PvP gear isn't always the best choice for PvE... I certainly wouldn't tank in most of it... but the weapons are almost always just as good for PvE as PvP, it's almost unavoidable that they would be.
At this point, you may well be 70 if you've done all of the quests in Blade's Edge and have run CoT: Old Hillsbrad and Sethekk Halls a few times. My advice would be to make up the difference in those instances and head into Netherstorm or Shadowmoon at 70 if at all possible. However, if you find yourself hard up for groups, you can head into both or either of those zones and do several quests to make up the difference. I'd recommend Netherstorm because it has, in my opinion, more quests that are easy to access if you don't have a flying mount yet. Area 52 in Netherstorm is faction neutral, and from there a level 69 player should be able to find plenty of quests that will take her or him to 70. Both Aldor and Scryer will have quest chains that begin in Area 52 at the inn: start them as soon as you can, and by the end you'll have received several good rewards from a nice DPS ring for Aldor or even a heck of a nice set of gloves for Scryers.
Hopefully when you hit 70, you'll have completed Old Hillsbrad and Sethekk Halls already, and thus be able to enter Shadow Labyrinth (a group with one keyed player is enough) and the Black Morass. In addition, you'll hopefully have the gold to purchase a flying mount (or will achieve it soon by doing the quests in Netherstorm and Shadowmoon) and can start running the various Tempest Keep 5-mans, as well as Shattered Halls in Hellfire Citadel and Steam Vaults in Coilfang. Since Patch 2.3, you should also be honored with pretty much every faction you'd need to get your heroic keys except perhaps the Sha'tar, which you can get fairly easily by a few Tempest Keep runs.
My advice would be to run every instance on normal mode a few times before doing the heroics, especially if you intend to tank them: there's good gear to get prepared for running heroics and Karazhan available to you here.
Next week, in fact, we'll start looking at getting ready for Heroics and Karazhan for both tanking and DPS warriors, or we'll look at the new abilities for warriors between 60 and 70 since we didn't have room to cover that here.