From Bulbapedia, the community-driven Pokémon encyclopedia.
When you awaken it's time to head to the Hill of the Ancients in the Great Canyon. You and your partner will have a conversation outside your base before deciding to depart and find out what happened to you.
Note: If you have done no side missions to level up yet, now is a great time to do so. Once you complete Great Canyon you will not be able to do missions or access the town for awhile, although you still will be able to train. If you want to be fairly comfortable try to level up to somewhere around level 15-20.
Great Canyon has 13 floors with Xatu on the 13th. You will not have to fight Xatu.
After you reach Xatu the story will continue. Xatu will tell you that you becoming a Pokémon is related to the disasters. This just deepens the mystery of you becoming a Pokémon, but then Xatu tells you he feels the world's destruction coming closer everyday. Once you see Gengar dropping in you will be asked to save and then a long journey will begin once you head to the Town Square.
Curse of Ninetales
Go to Whiscash to hear about the Curse of Ninetales. When finished, head back to your base where you will talk with your partner then go to sleep and meet Gardevoir in your dreams. After you awaken head to Town Square for the story to continue on.
You are now a fugitive and your Pokémon are running for their lives. Luckily as everyone else is catching up to you, you find a dungeon out of the way and decide to enter it.
(No matter how long you take on this Dungeon, no one will catch you)
Lapis Cave Pokémon
(You won't be able to recruit anyone on your first visit)
This dungeon is only 14 floors with no boss awaiting you at the end. It shouldn't be too much trouble for your team.
Before entering Mt. Blaze there is a save point that allows you to also access your storage. When you walk up to the cavern you are given the choice of Mt. Blaze or Rock Path. Rock Path is a 4 level dungeon implemented for those who did not level up before now. If your level is too low for Mt. Blaze you can run Rock Path a few times to level up and gather a few items before entering Mt. Blaze. When you feel you are fully prepared, choose Mt. Blaze.
Mt. Blaze is a total of 15 Floors. Once you pass the 12th Floor there is a save point waiting for you before the final 3 Floors.
Now there will be a save point, be sure to save.
Mt. Blaze Peak
Mt. Blaze Peak has three floors with Moltres waiting for you on the third. You can not recruit Moltres until after the end credits.