The Argent Tournament Grounds are located in the northeast part of Icecrown, just north of Sindragosa's Fall. You can pick up a quest that will lead you there at Krasus' Landing in Dalaran, and it's only a short flight north to the Tournament Grounds.
The Argent Tournament Grounds is home to a series of new quests, including dailies that you'll complete in order to gain renown with your side's (Horde or Alliance) racial factions. And yes, there are new achievements for rising up the Argent Tournament ranks. Your eventual goal is to become a Champion of the Horde or Champion of the Alliance, which will take a while. As an Orc, for example, you will start off by championing the Orgrimmar faction, but you will eventually be able to champion the other allied races. Finally, a new way to gain reputation with the capital cities!

Okay, mounted combat is fun. Anyone think we'll have aerial mounted combat with breath weapons in a future expansion?
Getting to know your way around the Argent Tournament Grounds can be worthwhile. In addition to relative proximity to Dalaran, the new hub offers all sorts of vendors, profession trainers and a flight path, making it at least as useful (and convenient) a place to visit as the Isle of Quel'Danas was in The Burning Crusade. You'll want to find the western pavilion, where you can meet Justicar Mariel Trueheart, who will start you on the road towards winning the Argent Tournament.
Depending on your faction, you'll start a series of quests that will train you in the new gameplay gimmick that the Argent Tournament is focused on: mounted combat. You'll get one quest for three of the basic combat abilities: Charge, Shield-breaker, and Thrust. Outside of your faction's pavilion, you'll hop onto a mount conveniently parked outside for you, a Sunreaver Hawkstrider for the Horde or Quel'dorei Steed for the Alliance.
Mounted combat is relatively simple, but strategy and skill is involved. Thrust is your melee attack, Charge zooms you towards your target from range, and Shield Breaker is a lance-hurling attack from afar that breaks down defenses. Defend is an activated ability that is at the core of staying alive in mounted combat. When you activate Defend, a shield icon appears around your character. Press it again for two shields, and three times for a third and final barrier. Each shield drops damage received by 30%, so it's key to keep your defenses up while breaking down your opponent's.