I've somehow managed to ruin my reputation with the Steamwheedle goblins, and they don't talk to me anymore. What should I do to quickly and easily regain their support?
I would like to point out a faster (and as yet unmentioned) way to get Steamwheedle rep. Specifically, grinding Southsea Freebooters (just south of Ratchet) and Wastewander (and/or Southsea Pirates*) in Tanaris. I find the rep/hour is much better than grinding shackle keys in Diremaul -- at least for me, who averages just over 2 keys/hour in DM (despite killing all 5 guards/captains 5 times an hour). And this can actually be done all the way to Exalted, if you are smart about when you grind what.
The rest of the guide details the order in which to farm reputations to get the best bonuses possible, and also includes many alternate methods you may find useful.
As of patch 4.2 on August 18, 2011, the answers listed here are largely out-of-date; Dire Maul no longer offers a way of repairing Steamwheedle rep, and many of the linked guides have not been updated to reflect changes from Cataclysm.
The fastest way to repair rep with the Steamwheedle Cartel is to do the "Bloodsail Forgiveness" quests with the goblins standing outside each of the towns. These quests are Making Amends outside Everlook (requiring 40 Runecloth and 4 Coal with each turn-in), Mending Old Wounds near Ratchet (requiring 40 Linen Cloth and 4 Crystal Vials per turn-in), Traitor to the Bloodsail outside Booty Bay (requiring 40 Silk Cloth and 4 Red Dye), and finally War at Sea outside Gadgetzan (requiring 40 Mageweave Cloth and 4 Strong Flux).
Each quest contributes 500 rep to the adjacent city and 250 to the other three cities. This means that, starting from maximum hatred, you can complete a specific quest 84 times (requiring 168 stacks of relevant cloth and 336 of the extra material) to raise your reputation to neutral with that city; the next quest can be exhausted after 42 turn-ins (requiring 84 stacks of cloth and 168 materials). The third quest is finished in a mere 21 attempts (requiring 42 stacks and 84 materials), while the fourth quest is done after just 11 turn-ins (requiring 22 stacks and 44 materials). Each of these quests is no longer available once your rep with the associated city becomes neutral, so don't over-stock on materials!
If you want to minimize your cost, you should complete the quests in this order: Ratchet, then Booty Bay, then Gadgetzan, then Everlook. This is the most cost-effective way of achieving neutrality in the fastest possible way. However, if you want to subsequently raise your Steamwheedle rep to exalted (whatever the reason), you'll actually want to do Everlook first, followed by Ratchet, then Booty Bay, then Gadgetzan. The reason for this is because the spillover rep from subsequent quests will continue to raise your rep with the earlier cities, and the location of farm-worthy pirates is such that Everlook (and, to a lesser degree, Ratchet) are considerably harder to raise than Booty Bay or Gadgetzan.
If you don't want to buy the cloth outright, the next best option is to farm pirates in one of three locations: in Lost Rigger Cover in Tanaris, on the three ships south of Booty Bay in Stranglethorn Vale, or on the ship that appears in Booty Bay proper during Bloodsail attack on the city. The pirates in Tanaris and Stranglethorn will drop cloth that can be turned in outside their city in addition to the rep you gain from outright kills, but it's still a much longer process than just buying everything you need outright.
If you're trying to get rep past neutral, though, those three farming locations are really your only options. They each come with their own sets of advantages and disadvantages. A particular note about farming: when one or more other players is also in the area killing mobs, the respawn rate will try to increase to match the rate they're being killed at. In general, if you can get in a group with somebody else killing guys in the same spot as you, you'll have much higher rep-per-hour gains.
- Lost Rigger Cove has lots of pirates in fairly high density spread over a large area; additionally, there's a pseudo-rare spawn on the south-easternmost ship that will award considerably more rep than pirates or even named pirate NPCs; however, he has a long respawn timer. Pirates here award ~5 rep toward Gadgetzan and ~2.5 rep toward the other cities.
- The ships south of Booty Bay have a very high density of pirates and named pirate leader NPCs that can award a lot of rep pretty quickly, but their respawn rate is just low enough to be frustrating; the presence of another player on the opposite faction will particularly exacerbate this. The rank-and-file pirates here award about 5 rep to Booty Bay and about 2.5 rep to the other three.
- The ship that spawns in Booty Bay has a lot of pirates and a very high respawn rate, but having Booty Bay's normal services and functions phased out can be kind of a hassle. There's a lot of chaos here, and I actually died on more than one occasion as a level 85 character just because I ran out of mana and didn't notice that my health had gotten so low. The other benefit of this method is that, like all Steamwheedle quests, doing the quest chain to start the phased event will award a little bit of rep all by itself. Low-level pirates here award about 5 rep to Booty Bay and about 2.5 to the rest of the Cartel.
You should probably try each location and determine which one works best for you. It's important to note that killing mobs for rep in one location alone cannot get you to exalted with the associated city; when I was doing it, I farmed Lost Rigger Cove until I stopped getting Gadgetzan rep, moved to the Bloodsail ships and farmed until exalted with Gadgetzan, and then moved back to Tanaris to get exalted with the other three factions.
Lastly, as I mentioned briefly above, any quests you can get in the various Steamwheedle cities will award a little bit of rep as well, but you'll have to be at least neutral or better to even have the opportunity to take them. Since this method isn't repeatable, and many people will have already done these quests prior to this situation, it's not as important as the methods above.
Besides killing thousands of Pirates you can spend 20Mageweave Cloth + 4 Strong Flux for 500rep each time until Neutral. The Npc is in Tanaris north of Gadgetzan.
The short answer is that the most efficient way to do this is killing some Bloodsail Buccaneers NPC.
There are more ways (Diremaul grinding, repetable quests) but they are more tedious.
A full explanation of all possiblities are detailed on WoWWiki