This game has a run page on SDA!

General speed tips

• Focus run everywhere. There's no excuse not to.
• Make "Open Palm" decisions at the end of the game. You will spare yourself from fighting extra fights.
• Remember that both the Start and A buttons can be used to skip dialogue.

Combat

• Fights will be quicker if you press [A A], [A A], [A A] instead of [A A A], [A A A]. Hopefully that made sense.
• Use Fortune's Favorite instead of Golden Star. The sword can hit many enemies simultaneously, and your enemies don't go flying if you use a power attack.
• Switching to a demon will remove all adverse effects (Disoriented, Immolated, etc.) but it takes lots of time, so try your hardest not to get burnt or whatever!

Specific numbers

• Buying Storm Dragon from Spear Catches leaf in the Forest costs 4,500 silver.
• Skipping the fight with the Lotus Assassin Slave Drivers requires 40 Intuition.
• Skipping the fight with Death's Hand the second time requires 70 Intuition.

Glitches

As far as we know, no one has discovered any useful "major" glitches -- meaning, those on the scale of skipping significant parts of the game. This holds true for the Xbox release; I really don't know about the PC version. The PC version might be easier to exploit because there are more characters/features/stuff in general to play with.

GameFAQs lists one sortof useful glitch (here). The only problem is that there are very few opportunities in the game to use this.

Instant Kill for Battles With Cutscenes

This glitch will instantly and completely kill whatever enemy you face if a cutscene is known to occur in the middle of the battle. To achieve this you must have a transformation style in your hot keys for styles. Then right before the cutscene is known to occur, transform. During the cutscene, you will notice that you have changed back to your normal form, but are blinking oddly. When you enter battle again, you will slide along the floor without walking, and will not be able to attack. Now use the transformation style again and the enemy will die, instantly.