FFXII - Walkthrough - Royal Palace
Open the urn; you obtain the map to the entire palace. Save your game.
In the next area, head north. Trigger some dialogue. Press Square, and when the guard approaches Vaan, easily slip by him. Now you're up to the "Lower Halls".
From the map, you can see there are 4 horizontal (east-west) hallways, and 2 vertical (north-south). You begin at the 3rd from the top; the signet is in the middle of the 2nd hallway, but guards abound, so you need to strategically call them over (using Square). However, there are guards at nearly every intersection, so you need to be careful how you go about doing it.
Don't worry, though - if you get "caught", you only start again at the entrance of the "Lower Halls", so it's no major punishment.
Three steps:
(1) Go to the intersection of the 4th horizontal/2nd vertical hallways. Press Square. The Soldiers from the 3rd/2nd will run to it... obviously, get away before they reach you.
(2) Go to 3rd/1st. Press Square. The ones from 2nd/1st will approach you. Go east to 3rd/2nd, which is now abandoned
(3) From 3rd/2nd, move up to 2nd/2nd, then into the middle; the Lion Signet is activated.
Now you want to go to the far west of the 1st horizontal hallway. So, go slightly west to 2nd/1st, call the guard over, and then do a counterclockwise run around the hallways; from 2nd/1st to 2nd/2nd to 1st/2nd and keep going west until you reach the end. You'll notice some a faint light glowing. Enter!
Head straight ahead; at the end of the hallway, on the left side, you'll notice a "Switch". Press it. The door to the treasure room can now be entered!
Cutscenes; you're introduced to Balthier and Fran, and obtain the Goddess's Magicite, an accessory with some pros (great defence) and cons (reduces your MP to 0. Yikes!)
In the Garden Stairs area: Run up the stairs... surely you can't get lost here.