Undercity is, well, confusing! It took me a long, long time to figure it out. After a while though, Undercity, like all other cities, starts to develop it's own pattern. To find Undercity, you first need to find Ruins of Lordaeron and navigate it's courtyard and entrance way. It's not hard, there is only one way to do it. You'll know you've found the right place when you see two undercity guards around a doorway, the elevator.
The elevator lets you off into a circular room. At its center, and down one set of stairs, is the bank. Around the top row, you find the Inn Keeper, barber, and flight master, as well as various other vendors, for weapons, armor and misc. items.
Around the bank on the lower level, you find guild masters and cooking trainers, one on either side. Notice the pattern of steps and how between each there is an entrance to the second ring.
The second ring holds only auctioneers. Not much to say about that.
Continuing on to the third ring, and this is where it gets tricky. I'll be jumping back and forth between what I consider third and forth ring. As you can see by your map, blizzard has divided undercity into quarters. Each quarter has the river running through it. For simplicities sake, I consider the other side of the river ring 4.
So, starting on the northern connection and heading clockwise we enter the Magic Quarter. In magic quarter ring 3 you have trainers for tailoring and vendors for warlocks and mages (cloth armor and wand vendors). In magic quarter ring 4 are their trainers (warlocks and mages), but not much else.
The next quarter is rouge quarters. Here, in the 3rd ring, are the first aid, leather work and skinning trainer, as well as their vendors, various rouge centric vendors and the bag vendor. In rogue quarter, ring 4, you have the rouge trainers. Finding the pattern yet? The Engineer trainer is also in this area.
The next quarter is the Apothecary with Enchanting, Inscription and Herbalism trainers and vendors in ring 3. Ring 4 is a bit of a maze. Downstairs you find the Alchemy trainer and mainfloor is the alchemy supplies vendor. Yes, you do hear crying. This area is a big gruesome.
In the center of Apothecary ring 4 is a raised area which leads to a long tunnel leading to the battlemasters area.
Last area, the War Quarter, ring 3, finds us with the mining trainer and blacksmith trainer, and their vendors, as well as various ranged and fist weapons vendors. Ring 4 has a mage, priest and warrior trainer.
As you can see, no Druid trainer here in Undercity. This is much the same as Orgrimmar. Only Thunder Bluff has a Druid trainer because only Tauren are able to roll a Druid. For some reason there is also no hunter trainer here. Possibly undead cannot be hunters.
I think that's all for the tour. If there's something I've missed, let me know please.