The first thing you’re going to want to do is to take off all your armor and weapons.
From Silvermoon City, find the nearest road and begin following it generally south. (If the road you picked curves AWAY from the south, you may need to run overland to another road.) At some point a monster will begin attacking you, at which point you can either kill it or just run past. Eventually you should arrive in the Ghostlands, which have a long road from the north to the south which you should follow. The south end of this road connects to the Eastern Plaguelands; your objective here is to follow the circuitous path from the Ghostlands area transition around to the bridge to the Western Plaguelands on the south-west border of the region. If you don’t know the way, it’s relatively easy to just always turn right whenever you get to a fork in the road; doing this from the beginning of your entry into the Plaguelands will eventually bring you to your destination.
(At some point before you found your way to the Western Plaguelands, you almost certainly died. Do not panic: this is normal. You should be standing in a graveyard. If you look on your minimap, there will usually be a red arrow pointing the way back to your corpse. You can either follow this arrow toward the gravestone icon and respawn at your body, or you can simply continue your journey toward Orgrimmar. I recommend the latter course of action: you’ll find your journey far easier, but you’ll have to tolerate a period of statistical diminishment at your destination and, depending on whether you followed my advice about taking off your equipment, you may also have a minor repair cost. If you choose to respawn, you run the risk of just dying again, adding considerable time and energy to your trip. On the other hand, running around as a corporeal body gives you the opportunity to unlock new flight points you run past, as well as earn XP by entering areas you’ve never seen before. I’m not sure whether the XP rewards are also available as a spirit.)
Continuing from where I left off: you should now be arriving in the Western Plaguelands, preferably as a ghost. Point yourself generally toward the west and hold down the W key (or whatever your keybinding for ‘walk forward’ is). Assuming you take the time to jump over or move around obstacles as they approach your character, you will wind up in one of several locations: Eversong Woods, Hillsbrad Foothills, or Silverpine Forest. If you find yourself in Hillsbrad Foothills, follow the path south and to the west until you get to Silverpine. Once you’re in Silverpine, walk north toward Eversong Woods—and if you arrived in Silverpine directly from the Western Plaguelands, you’re probably in the lake, so drag yourself ashore and strike out northward.
Once you’re in Eversong Woods (however you managed to get here), hit M—or whatever key your map is bound to. On the map, draw an imaginary line between yourself and the middle of the map; this is the approximate heading you’ll want to use. Face in that direction and begin walking; travel here should be no more difficult than it was around Silvermoon. You’re looking for the Undercity, or more specifically, Brill; you can find them more easily if you enable “Points of Interest” on your minimap. Brill’s just to the north of Undercity; between the two locations is a pair of matched towers. Climb one of them and wait for an airship to arrive. Walk on board.
Depending on which tower you climbed, you'll either find your airship deposits you in Orgrimmar (in which case, congratulations; click Return to Graveyard and revive yourself, which should bring you back inside the city—or at worst, in the town of Razor Hill, immediately south of the city) or Howling Fjord (in which case, just wait on the airship until it returns to Brill, then descend that tower and climb the other one).
I hope this helps!