It's time to return to Turnback Cave and the Distortion World to catch Giratina if you didn't already. If you didn't catch Giratina yet, you'll want to take about 40 Dusk Balls with you. Either way, there is an important item to pick up.
Use Defog to remove the fog. Read the pillar that says "...Past three pillars...to the sleeping... ...before 30 is surpassed...". It means that to access the room with a portal to the Distortion World, you must find 3 rooms with a pillar before going through 30 rooms. This cave is highly unusual in that the layout of rooms is randomized each time you visit. Inside the maze, for a hint on which doorway to take, study the breakable rocks and go through the doorway that the odd pattern points to. The further you go, the stronger the wild Pokémon will be.
Once you've gotten past the three pillars and got whichever item was waiting for you, you'll find the room with the portal to the Distortion World. If you failed to catch Giratina last time, you'll find it guarding the portal. This time, it's in Altered Forme.
One slight advantage if you decided to KO Giratina in the Distortion World earlier is that in Altered Forme, it prioritizes defenses over offense, so its attacks are not as strong. Its moveset is still the same, though. Save before facing it; otherwise you'll have to defeat the Elite Four again to get another chance to catch it. If it does come to that, rechallenge the Elite Four before completing the mission at Stark Mountain. Otherwise, their teams' levels are increased, making it harder to defeat them.
In this battle, use the same tactics as you did when facing Azelf and Uxie and you'll be fine.
When ready, use the portal to return to the Distortion World.
This time around, you do not have to worry about getting lost. Follow the path until you find the Griseous Orb. The Griseous Orb, when held by Giratina, will allow it to maintain it's Origin Forme outside of the Distortion World.
When finished in the Distortion World, use the portal to return to Turnback Cave. Any doorway from the portal room will take you back to the entrance room.
Your next destination is Canalave City. Enter Sailor Eldritch's house in the southwestern area of town. You'll see that his son is trapped in a nightmare. Talk to his mother and you'll learn that you need to find an item called the Lunar Wing on Fullmoon Island to save him. Talk to Sailor Eldritch and he will take you to Fullmoon Island.
Follow the path and head into the forest clearing where Cresselia is. When you interact with her, she will flee, but she leaves the Lunar Wing behind. Return to Eldritch and have him take you back to Canalave City, then return to the house and use the Lunar Wing to heal his son. As for Cresselia, she's similar to Mesprit, Zapdos, Articuno, and Moltres— she is roaming around the Sinnoh region so you need to use the Marking Map to track her.