You enter Outland through the Dark Portal in The Blasted Lands
There are NO class trainers in Outland
There is NO Auction House in Outland
There are NO Grand Master profession trainers on the old continents; they are all in Outland
The Alliance trainers for 300-375 are at Honor Hold (unless otherwise noted):
Jewelcrafting - Tatiana is just inside the Inn near the front door
Skinning - Jelena Nightsky is in the main room of the Inn, by the stairs
Tailoring - Hama is upstairs on the second floor of the Inn
Cooking - Chef Gaston is in the back room of the Inn, near the cooking fire
Leatherworking - Outside the Inn on the left wall, by the outhouses.
Alchemy - Inside the mage tower on the bottom floor
Herbalism - Inside the mage tower on the bottom floor
Enchanting - Inside the mage tower on the upper floor
Engineering - Lebowski is outside the Keep, on the right hand side
Mining - Hurnok Grimmord is inside the smithy
Blacksmithing - Humphry is inside the smithy
First Aid - Burko at the Temple of Telhamat, on the left side of the second flight of steps
Fishing - Just outside of the Cenarion Refuge in Zangarmarsh
The Horde trainers for 300-375 are at Thrallmar (unless otherwise noted):
Jewelcrafting - Kalaen is inside the Inn
Skinning - Moorutu is outside under a small tent in front of the Inn
Leatherworking - Barim Spilthoof is with the skinning trainer :)
Tailoring - Dalinna is inside the Inn
Cooking - Baxter is inside the Inn
Alchemy - Apothecary Antonivich is inside the mage tower on the west side of Thrallmar
Enchanting - Felannia is inside the mage tower on the west side of Thrallmar
Engineering - Mack Diver at Zabra'jin in Zangarmarsh, located just outside the fence on the southeast side of town, standing under the bridge that leads into town.
Mining - Krugosh is standing in front of the barracks
Blacksmithing - Rohok is at the forge just south of the mage tower
First Aid - Aresella is at Falcon Watch on the southwestern side of Hellfire Peninsula
Fishing - Zurjaya at Zabra'jin in Zangarmarsh, standing by the bonfire in the center of town.
If for some reason you are unable to find your trainer in your starting town, there is a full complement of Grand Master trainers located in the Lower City area of Shattrath City. You can ask a guard to mark your mini-map if you need help locating them.
The nearest bank is in Shattrath City, which you can reach by going west through Hellfire Peninsula into Zangarmarsh, then follow the signs to Shattrath City.
Lieutenant Amadi is at the small tower just northwest of Honor Hold
Altumus is behind a broken wagon on the road leading east from Honor Hold
Bonechewer Orcs are just north of Honor Hold, along the main east/west road.
Salvaged metal and wood are found on the ground near the Bonechewer Orcs
Expedition Point is directly east of Honor Hold, along the edge of the zone.
Zeth'Gor is in the southeast corner of the zone.
Cursed Talismans are dropped by Bleeding Hollow Orcs in Zeth'Gor
Horde Blade Throwers can be found on the north side of the Path of Glory (main road going east/west through the zone).
Trampled skeletons are on the Path of Glory
Stonescythe Alphas and Whelps are in the Great Fissure, near Falcon Watch in the southwestern part of the zone.
Sedai can be found behind the flightmaster at the Temple of Telhamat, yes you'll need to jump :P
Flight paths in Hellfire Peninsula are at Honor Hold and the Temple of Telhamat for Alliance. Horde towns are Falcon Watch and Thrallmar (with FPs I hope).
Follow the road west from Honor Hold to enter the next zone: Zangarmarsh.
There is no flight path at the Cenarion Refuge in Zangarmarsh, it's just a quest/faction hub. Take the road north, then follow the west fork from there to the next town (Telredor) for the Alliance flight path. Horde should take the east fork to find their town of Swamprat Post, they also have a town at Zabra'jin in the west.
Glowcaps and Bog Lord Tendrils are faction turn-ins for Sporeggar, Unidentified Plant Parts are faction turn-ins for Cenarion Refuge.
Save everything that's not gray - odds are you'll find a quest that requires it. If not, there's a good chance you can sell it to other players on the Auction house (for quests you haven't found yet).
To reach Shattrath City, follow the road south from Cenarion Refuge into Terokkar Forest. Shattrath City is right there after you enter the zone.